Sources said the Bears have included Bennett in trade talks since discussing a potential deal with the Seattle Seahawks, who ultimately acquired tight end Jimmy Graham in a trade with the New Orleans Saints.
The Bears, who declined comment, are open to trading the 28-year-old Bennett because he's seeking a new contract and does not plan to show up in camp with Chicago, or anywhere else, without a new one, the sources said.
Any team trading for Bennett would probably want to sign him to an extension beyond the two years he has remaining on his current contract.
Bennett is coming off a career year in 2014 in which he topped all NFL tight ends in catches (90) and also set personal bests in receiving yards (1,038) and touchdowns (six).
He was a late addition to the Pro Bowl roster after New England's Rob Gronkowski pulled out following the Patriots' Super Bowl victory, making Bennett the first Chicago tight end selected to the Pro Bowl since Mike Ditka in 1965.
Bennett signed a four-year, $20.4 million deal with $9.215 million in guarantees on March 13, 2013. His compensation for 2015 is scheduled to include a $4.9 million base salary and $100,000 workout bonus.
Since joining the Bears, Bennett has 155 catches for 1,675 yards and 11 touchdowns.
Jeff Dickerson of ESPN.com contributed to this report.