A source told ESPN's Adam Schefter on Friday that the Bears are working with Bennett's agent, Kennard McGuire, to find a trade partner.
The 6-foot-6, 273-pound Bennett is entering the final year of his current contract, with $5.085 million owed to him next season.
The former Pro Bowl tight end boycotted the entire voluntary offseason program in 2015 in search of a new deal that never materialized.
Bears general manager Ryan Pace and head coach John Fox have both deflected questions about Bennett's future in Chicago.
"I talked to him right at the end of the season," Pace said at the NFL Scouting Combine in Indianapolis last week. "I haven't talked to him recently. Really with him, he's under contract. We have a lot of players that aren't under contract, so that's where our focus is right now."
Bennett, who turns 29 next week, publicly questioned his role last season and went on injured reserve Dec. 8 with fractured ribs.
Bennett, in three seasons with the Bears and eight overall in the NFL, has 348 career receptions for 3,586 yards and 23 touchdowns in 119 games with the Dallas Cowboys (2008-11), New York Giants (2012) and Chicago (2013-15). He has started 41 of 43 games with the Bears and totaled 208 catches for 2,114 yards and 14 touchdowns.
Bennett finished last season with 53 catches for 439 yards and three touchdowns in 11 games.