He has been speaking with several teams, but cannot officially sign a deal until July 5.
Lecavalier is believed to be looking for a five-year contract, worth around $5 million per season.
The Dallas Stars, Philadelphia Flyers, Boston Bruins, Florida Panthers, and Ottawa Senators are among the teams very interested in Lecavalier, along with Montreal, but it's unclear which team is willing to offer five-years.
The Detroit Red Wings, Calgary Flames, Anaheim Ducks and Nashville Predators were also interested in Lecavalier, as were several other teams, but TFP has learned that all four of these clubs are now out of the race.
Canadiens GM Marc Bergevin is believed to be only interested in signing Lecavalier to a three-year agreement. It's unknown if talks will continue.
In 39 games with the Lightning this past season, Lecavalier registered 10 goals and 22 assists for 32 points.
The former first-overall selection in the 1998 NHL Entry Draft, Lecavalier has played in 1,037 career NHL regular-season games, all with the Lightning, and has accumulated 383 goals and 491 assists for 874 points. He helped lead the Lightning to their first Stanley Cup championship in 2003-04.