Nunez suffered a knee injury last season while playing for Boston, but has been medically cleared.
At least eight teams -- including the Yankees and Blue Jays -- have been in contact with Nunez.
Aside from Nunez, the Mets have reportedly expressed interest in Pirates INF/OF Josh Harrison, and free agent infielders Todd Frazier and Neil Walker.
Nunez, 30, hit .313/.341/.460 with 12 HR in 114 games last season with the Giants and Red Sox, while getting time at second base, third base, shortstop, left field, and right field.
He has been a negative defender at second base the last two seasons, per FanGraphs.
Mets GM Sandy Alderson has suggested multiple times this offseason that the team would rather acquire someone who can play second base (instead of a Todd Frazier-type who can only play third base and first base), in order to keep Asdrubal Cabrera at third base -- where he grades out better defensively.
Cabrera was worth -6 DRS (defensive runs saved) in 274.1 innings at second base last season, and was worth 1 DRS in 350.1 innings at third base.