Applebaum says union president Yves Francoeur suggested he would have problems if he didn't stop a plan to end the current three-day work week for cops.
Francoeur denies the allegations.
Applebaum says Francoeur made the threat in an email and a phone conversation, giving the mayor a deadline to get involved.
The mayor says he interpreted the warning as a threat to his reputation and his integrity and insists he won't be swayed.
Under an experiment, Montreal police are working a three-day week but that is due to change in March.
Francoeur says he would never make threats and that he deals regularly with government officials.
He says Applebaum is trying to deflect attention from a raid on city hall this week by the province's anti-corruption squad on Tuesday.
Applebaum met with detectives but says he is not personally under investigation.
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