There is no suggestion of any criminality on Redknapp's part, or that he knew the opposition players were betting, or that he was aware his own players had bet on the match when the game kicked off, the newspaper said on Friday.
Redknapp, 69, has managed West Ham, Portsmouth, Southampton, Tottenham and Queens Park Rangers.
The Telegraph said it was not identifying the match in question, the teams involved or the players concerned for legal reasons.
This is the latest in a series of Telegraph stories into alleged corruption in English football.
Sam Allardyce left his job as manager of England on Tuesday after he was filmed discussing ways to circumvent transfer rules with undercover reporters.