So you have two characters who seem to work well together and they seem to have a decent relationship going on. They are loyal to one another and seem to respect each other very well. You can tell that they're friends, right?
Well, in the case of one particular character, not exactly.
This is the trope where a character flat out denies his friendship with one or another for some reason. Perhaps the character is either The Lancer who has trouble getting along with The Hero and doesn't like working with him that much. He may be a natural-born loner who isn't concerned about intimate friendship. Or he's just a prideful Jerkass who thinks that having friends makes him uncool and less of a badass. But note that in spite of all of these reasons, that doesn't mean the person wants to wish death on him. When the chips are down, he still does help out his 'friend' because it's the right thing to do.
Sometimes, other versions of this trope would be that the character may not be in denial after all. Reasons being that the character doesn't fully know the hero since in order to be a friend, you must be intimate to know who they are; most examples of this would have a Nice Guy call the other "buddy", "pal", or "friend", but the offended character responds with "I'm not your buddy/pal/friend." A character in the workforce is appointed to work with a group of people, but friendship isn't a part of his job description since Being Personal Isn't Professional. Even to the extremes, the character in question is actually the hero's arch-nemesis either by honesty or revealance. Obvious, isn't it?
On a brighter note, there are high chances for the character to consider that the other character is truly their friend, and from that point on, the bond will become stronger and better than it was before. If they are not technically friends, they will soon become closer as the series progresses.
This is often common in crime scenes where a character denies friendship with the second party as a ruse to escape responsibility for his crimes, especially in a trial.
This is the character trait of one or two characters in duo of Vitriolic Best Buds. A group of characters can do this to one particular person in their group, which is The Friend Nobody Likes. Can be the kind of relationship the Ragtag Bunch of Misfits would have. Often related to Hidden Heart of Gold. Can be utilized as the tool to permanently end a friendship, which is We Used to Be Friends, or at least temporarily in a Feud Episode.
Compare She's Not My Girlfriend and Just Friends for the more romantic versions.
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Anime & Manga
In the first season of Sonic X, Knuckles would occasionally deny being friends with Sonic (as the page quote above tells you), but this is due to his disapproval of the hedgehog's reckless nature and snarky attitude. It also doesn't help that Knuckles is quite jealous of Sonic's freedom and popularity despite being a loner with a task to guard the Master Emerald.
This happens in Dragon Ball Z when Piccolo became infuriated of being defeated by the androids, but the others try to calm him down:
Krillin: "Wh-what's that look on your face? Do you have a plan?! C'mon, tell us! We're your friends!"
Noir. Mirielle needed Kirika to stay alive to help her find the answers to Soldats, but this doesn't change the fact that both are Heterosexual Life-Partners with a dose of Ho Yay.
In Bleach, Uryu Ishida will make it very clear that he and Ichigo are not best friends, despite being an frequent ally to him and the other Soul Reapers.
Lina Inverse and Naga the White Serpent in The Slayers OVAs and movies are Vitriolic Best Buds who will vehemently deny that they are best buds, or any kind of buds. Both claim that the other is their sidekick. Or rival, depending on the mood.
Jack of 48 Hours has this quote that comes in mind as he forms a grudging partnership with Reggie, a convict, to hunt down Ganz:
Jack: Now, get this! We ain't partners. We ain't brothers. And we ain't friends. I'm puttin' you down and keepin' you down until Ganz is locked up or dead. And if Ganz gets away, you're gonna be sorry YOU ever MET me!
Turkey Creek Jack Johnson: "Why you doin' this, Doc?"
Doc Holliday: "Because Wyatt Earp is my friend."
Turkey Creek Jack Johnson: "Friend? Hell, I got lots of friends."
Doc Holliday: "...I don't."
In Pineapple Express, Dale and Saul fall into this trope after Dale got himself arrested for smoking a joint and selling weed to middle school kids, also escaping from the police. The two got into an argument, in which Dale tells Saul that they were never friends and never will be. The two did reconcile at the end of the movie, however.
In Baccano!, Chi tells Christopher that he doesn't consider him as a friend, but he considers him as somebody he wants to kill despite that he never gone through it due to circumstances.
In The Kite Runner, Hassan considers Amir a friend, but Amir considers him anything but. This is only because Hassan is Amir's servant as he points out to Assef. He's also a bit jealous of how his father treats Hassan.
The Cosby Show: A parental example has Cliff trying to tell Rudy that he's her friend, but Rudy responds that he's her dad.
The pilot episode of The West Wing:
Laurie: "Tell your friend P.O.T.U.S. he's got a funny name, and he should learn how to ride a bicycle."
Sam Seaborn: "I would, but he's not my friend; he's my boss."
The Buffy the Vampire Slayer episode "Lover's Walk" has Buffy telling Angel that they aren't friends and never will be just after telling him that she's not coming back.
And in The Blind Banker, when Sherlock introduces John to an acquaintance of his from university, Sebastian Wilkes, he says: "This is my friend, Dr John Watson," which John immediately corrects with "Colleague."
Charlie: "Yeah, I used to— And don't call me 'buddy'."
Merlin: Frequently between the titular character and Arthur.
Lead singer of Alice in Chains Layne Staley states in his final interview before his tragic death that he never bothered to call the rest of his AIC members to help him with his drug addiction and states that said members are not his friends.
The opening statements of "Birdhouse In Your Soul" by They Might Be Giants implies that they are not a friend in a traditional sense, but their job as the protector of its owner is the closest thing to one:
"I'm your only friend; I'm not your only friend, but I'm a little glowing friend, but really I'm not actually your friend, but I am."
This verse of "Automatic Stop" by The Strokes deal with the a guy and a girl who have ended their romantic relationship and realize that they aren't friends to begin with after knowing each other for a short period of time:
Wait, I'm gonna give it a break.
I'm not you friend,
I never was.
Rapper 50 Cent has a quote similar to the one from 48 Hours in both of his songs "Life's On The Line" and "Get In My Car".
"We ain't buddies, we ain't partners, and we damn sure ain't friends."
Before teaming up with The Rock in a match, "Stone Cold" Steve Austin states in an interview that both arch-rivals are partners for the main event and do have respect for one another, but as for the friends part, "Hell No!". The Rock was thinking the same way, too.
Religion and Mythology
Older Than Feudalism: This trope was prophesied in the The Bible. Jesus tells Peter that he will deny Him three times before the rooster crows, but Peter assures Him that he will never do that to Him. However, as Jesus was arrested that night, somebody in the crowd asks Peter was he with Jesus. Trying to avoid getting himself into further trouble, Peter responds with "No! I do not know Him!". As the rooster crows and with Jesus looking at him, Peter realizes that he was wrong.
At the beginning of Wicked, Glinda has to deny her friendship with the recently deceased Elphaba for political reasons.
Falco has something to say to the lava birds in Star Fox 64:
Family Guy: In one episode Peter and Meg are forced to spend time together and realize that they actually like each other; but they agree that Peter will still act like a jerk to her when they're not alone.
Arthur: Arthur and Molly (one of the Tough Customers bullies) spend time and get along together when Molly's mom makes her let Arthur & DW use their backyard pool. They agree to act like they hate each other when there are other people around.
Hubie: "Hey buddy, looks like we're two birds of a feather."
Rocko: "Don't call me buddy!"
Hubie: "And, gee, it looks like I got me a friend!"
Fluttershy does this to Discord in My Little Pony: Friendship Is Magic episode "Keep Calm and Flutter On" after he starts causing havoc in Ponyville. This actually pivots his Heel-Face Turn as he realizes he actually wants her friendship (being the first he's ever had due to his callous pranks), after which she is perfectly happy to take him in.
At the beginning of Ice Age, Manny denies being friends with either Sid or Diego. Eventually he comes around.
Animaniacs: An arrogant business man sitting next to the Warners on a plane declares that "I'd rather be eaten by cannibals than be your 'special friend'!" He soon comes to regret his words. Doubles as a Brick Joke when he escapes the plane to land on a desert island.
From the episode "Unjustly Neglected Drama" of The Life & Times of Tim, an actor is interrogated by a detective whether or not is he friends with Tim. He denies it. In fact, the denial was a line that was used from a play:
In an episode set on the first day of school, class nerd Martin greets class bully Nelson as an old friend, since they had spent their summer together at space camp; but Nelson tries to downplay their relationship.
In another episode, the kids of Springfield go to Shelbyville to find their missing lemon tree, and Nelson & Martin are paired up. When Martin gets in trouble, he calls upon Nelson for help:
Martin: No one manhandles the bosom chum of Nelson Muntz. Spring forth, burly protector, and save me!
*long pause; the kids look around*
Nelson: Aw, jeez. I never hang out with him, normally.
This was hilariously subverted in "Treehouse Of Horror XX":
Marge: *Dracula, the Mummy, Frankenstein's monster and the Wolfman are at the door* Friends of Homer's?
Homer: They're not my friends.
Count Dracula: We brought a bottle of booze. Booooze!
Homer: Come in, come in.
It is used to hide some aspects of an individual that aren't mainstream of the social group like hiding that the individual likes X series or that hobby that isn't that well viewed in society. Reasons vary but the most common in this tropers opinion is the hiding of a factor.
Real Madrid football player Xabi Alonso admitted that he has no friendship with fellow Real Madrid teammate Cristiano Ronaldo, but in the football field, there is, as he quotes, "appreciation" and "professional respect" for each other.