Gee, 28, started for the Mets on Opening Day last season. However, he was poised to be bounced from the rotation this year with Matt Harvey returning from Tommy John surgery. Zack Wheeler's loss for the upcoming season, also due to an elbow-ligament tear, reopened a rotation spot.
The Mets will start Bartolo Colon, Jacob deGrom and Harvey during a season-opening series against the Nationals in Washington. Jonathon Niese and then Gee will make their season debuts against the Atlanta Braves, as the Mets' opening road trip continues at Turner Field.
Gee acknowledged he has been told to prepare for the season's fifth game, but added: "Your guess is as good as mine right now. I don't put too much stock into things said anymore. I'm getting ready to start, obviously."
Gee has produced a 2.49 ERA in 21 2/3 Grapefruit League innings.
Despite Gee being poised to be named to the rotation, the Mets may trade him early this season, a source told ESPN.com.
Montero, who is expected to open the season in the bullpen with a potential late-inning role, is major league-ready. He just fired six scoreless Grapefruit League innings against the Miami Marlins on Monday and has a 2.40 spring training ERA.
The Mets also have a stockpile of talent in their Triple-A rotation, with Noah Syndergaard, Steven Matz and Matt Bowman.
Gee is 40-34 with a 3.91 ERA in 106 career appearances (103 starts) for the Mets.
He went 7-8 with a 4.00 ERA in 22 starts last season. He missed two months with a strained right lat muscle.
Gee is due to make $5.3 million this season. He is under the team's control through the 2016 season.