The Chicago Tribune first reported the Bears' interest in Maneri.
Maneri started eight games and played over 400 snaps for the Chiefs last season.
Maneri, a college teammate with Bears second-year tight end Evan Rodriguez at Temple University, did not receive an exclusive rights free agent tender from the Chiefs.
The Bears are looking to replace blocking tight end Matt Spaeth who signed a two-year contract with the Pittsburgh Steelers on Monday after visiting the Bengals last week. Another ex-Bears tight end Kellen Davis is now believed to be under consideration by the Bengals, who are clearly interested in adding a blocking tight end in free agency.