To make room, the Giants waived defensive end Devin Taylor and wide receiver Ed Eagan, and placed linebacker Keenan Robinson on injured reserve. Robinson injured a quadriceps against the Rams on Nov. 5.
The Titans made Ayers a second-round choice in the 2011 draft. He has played in 91 regular-season games, with 51 starters, for the Titans, Patriots, Rams and Colts. He also has played in three postseason games, including Super Bowl XLIX with New England.
The Giants also signed linebackers Kelvin Sheppard and Deontae Skinner the past two weeks after a rash of injures at the position. B.J. Goodson (ankle), Devon Kennard (quad) and Robinson were inactive Sunday against the 49ers, and Sheppard left with a groin strain.
The Rams made Greco a third-round pick in 2008. He has played 111 games for the Rams and Browns, with 45 starts at right guard, 24 at left guard and one at center.
The Giants also signed tight end Matt LaCosse and offensive lineman Nick Becton to their practice squad.