The mayor said he spoke on the phone Friday morning with Yves Francoeur, who told him if he didn't change his mind about police scheduling by 6 p.m. today, he would launch personal attacks against him.
Applebaum said Francoeur gave him an ultimatum to extend a police-scheduling pilot project to next year.
The project, which involves 3,000 police officers, allows them to work three days a week with extended hours.
Francoeur said it allows the police department to extend coverage on peak hours and does not overwork officers, unlike the shift rotations they are usually subject to.
"The objective was to avoid having police officers alternate shifts on days, evenings and nights. All recent studies show it, it's not good for your health," he said. "The new schedule is a benefit for everyone and I told the mayor this morning that it's a benefit for citizens too."
The mayor said it was decided not to continue with the program after consultations with police officials and the director of the city's human resources department.
Francoeur said he suggested the city keep the project going until January 2014. He said eight of the 13 months the pilot project was running were overshadowed by student protests in Montreal.
"I told him I want an answer before 6 p.m. this afternoon because if you don't tell them at the City of Montreal ... you wait for weeks and weeks," he said.
Francoeur said he has talked to his lawyers and wants Applebaum to retract the comments he made about being threatened.
Applebaum said Francoeur's ultimatum took things too far.
"He clearly stated that as a result of my decision, the police brotherhood would be on my back," he said.
Applebaum said he will not give in to threats and ultimatums.
"It's deplorable that a police officer openly threatens the mayor of Montreal."
Francoeur admitted that he told Applebaum that the union has a plan to respond if the pilot project is not extended, and that Applebaum will be held accountable.
In an interview with CBC Radio's Homerun, Applebaum said he would not bend to Francoeur's request.