It had been widely reported through numerous media outlets that the Canadiens and Buffalo Sabres were discussing a deal involving Briere.
Briere has recently been a healthy scratch for the Habs, but will be playing for Montreal tonight.
He has a no-movement clause in his contract, but it's believed he would be willing to waive it under the right conditions.
It's unclear if the Canadiens have approached Briere about specifically waiving his NMC, but Bergevin has spoken with the 36-year-old about his discontent with his lack of playing time.
Bergevin has been trying to acquire a top-line scorer since the summer, but has come up empty. He has been trying to free up some salary space, as well, dangling the likes of Brian Gionta and Rene Bourque.
Teams have inquired about such stars as P.K. Subban, Max Pacioretty and Brendan Gallagher, but it's highly unlikely the Canadiens move any one of them.
The Habs and Sabres have continued their discussions, and Bergevin is also maintaining conversations with other clubs, one of which is believed to be the Winnipeg Jets, as Montreal has shown interest in left wing Evander Kane.
In 876 career NHL regular-season games, to date, with the Phoenix Coyotes, Buffalo Sabres, Philadelphia Flyers and Canadiens, Briere has accumulated 291 goals and 378 assists for 669 points, along with 710 penalty minutes.
He was originally drafted 26th overall by the Coyotes in the 1996 NHL Entry Draft.