He had two-years left on his previous contract that had a salary cap hit of $6.5 million, though he was set to earn $3 million next season and $2 million in 2014-15.
Briere had spoken with six teams on Wednesday, but narrowed down his list to three clubs. He'll be continuing those discussions throughout Thursday before making a decision.
According to Pierre LeBrun of ESPN and TSN, the New Jersey Devils and Nashville Predators are the other two teams in the running for Briere.
In 34 games with the Flyers this season, Briere registered six goals and 10 assists for 16 points.
Meanwhile, the Canadiens are also working the trade wire, as GM Marc Bergevin is looking to add another goal-scorer and a top-four defenseman.
The Habs have had conversations with the St. Louis Blues about left wing David Perron and the Vancouver Canucks about defenseman Alexander Edler.