The move stems a tide of player departures for the Colts since they beat the Chicago Bears in the Super Bowl on Feb. 4.
On Tuesday, receiver Brandon Stokley signed with the Denver Broncos and defensive tackle Montae Reagor signed with the Philadelphia Eagles. Both were cut by the Colts to save money against the league's $109 million salary cap.
Two assistant coaches, Leslie Frazier and Diron Reynolds, also left for new teams.
Klecko is the second unrestricted free agent to sign with the Colts, joining linebacker Rob Morris.
Klecko had a hand in solidifying a run defense that was the NFL's worst during the regular season. He made a more noticeable impact on offense, where he doubled as a fullback late in the season. He caught a touchdown pass to help beat Miami in the regular-season finale and caught another to help rally the Colts from a 21-6 halftime in the AFC Championship Game victory over New England, his former team.
The Patriots selected Klecko in the fourth round of the 2003 draft. He caught three passes and made 14 tackles in 2004 before being placed on injured reserve and had eight tackles in 2005 while winning Super Bowl rings in 2003 and 2004.