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As far as I know, the police would need to subpoena the bank to access the account owner's information. They would be requesting information on a specific account or group of accounts that they have in connection to the crime. There are other banking regulations that may facilitate this, and banks are required to notify law enforcement of suspected activity like deposits over a specific amount or transactions with suspected terrorist channels.
A variant of No Warrant? No Problem!?
Several examples I've seen were written pre-fall of the USSR (and involved MI 5 and/or 6), so there might have been unstated automatically-granted access in the name of national security... but the traitor seemed genuinely surprised they'd have thought to check his bank account where large sums kept appearing, so clearly it's not something they do regularly.
Particularly frustrating is the corollary (no one is aware the character is receiving/giving large amounts of money): a lot of these involve traitors with gambling debts or such that allow the villains to bribe them, when you'd think the employers would know about such weaknesses considering the importance of secrecy.
Not really a trope, more of a Hollywood Law thing: crime series and spy shows often use bank accounts to prove or at least suspect that a character (living or posthumous) is guilty of something (often blackmail, whether they're blackmailing or being blackmailed, and bribes as well). There never seems to be any difficulty in doing this, but the examined person always reacts with outrage, so is it legal for the police to look into someone's bank account?