The source said that Briere and general manager Paul Holmgren met last week, at which time the Flyers forward was informed of the team's decision.
The move will shave $6.5 million off the Flyers' salary cap. Philadelphia had one of the NHL's most expensive payrolls last season and has been forced into a position to shed salary in order to get under next season's cap, which will drop from $70.2 million to $64.3 million.
Under NHL rules, teams have the option of two amnesty buyouts to terminate contracts before the start of the 2014-15 season.
The Flyers are still on the hook to pay Briere two-thirds -- about $3.3 million -- of the $5 million the player was owed as part of the eight-year, $52 million contract he signed in free agency on July 1, 2007.
Briere is a 15-year NHL veteran, who had six goals and 10 assists in 34 games last season. He's a seven-time 25-goal scorer, but his production has tailed off as a result of various injuries, including a concussion and a broken wrist.
The move does not come as a surprise, and was one Briere anticipated could happen in late April, after the Flyers missed the playoffs by finishing 10th in the Eastern Conference.
"I certainly hope I'll still be here," Briere had said. "But I also understand it's a business, so we'll see what happens with that."
The Flyers have already begun retooling their roster after agreeing to sign newly acquired defenseman Mark Streit to a four-year, $21 million contract. Philadelphia acquired Streit last week in a trade with the New York Islanders.
Overall, Briere has 286 goals and 659 points in 847 games split between the Flyers, Buffalo Sabres and Phoenix Coyotes. He's been even more productive in the playoffs, having scored 50 goals and 109 points in 108 games.
The 35-year-old emerged as an offensive star and leader during three-plus seasons in Buffalo, when he helped the team reach the Eastern Conference finals in both 2006 and '07. The Sabres lost the conference final series both times.
Briere helped lead a Philadelphia run to the Stanley Cup finals in 2010, but the Flyers missed the playoffs last season, and were eliminated in the second round in 2011 and 2012.
ESPN.com's Pierre LeBrun contributed to this report. Information from The Associated Press also was used.