Bennett, 26, experienced a breakout season in 2012, when he caught a career-high 55 passes for 626 yards and five touchdowns, in addition to showing consistency as a run blocker. The Giants made a late push to re-sign Bennett before the free-agency signing period, according to sources, but it's unknown whether the Bears could swoop in and land him.
The Chicago Sun-Times first reported the Bears' interest on Tuesday.
Selected in the second round of the 2008 draft by the Dallas Cowboys, Bennett signed a one-year contract worth $2.5 million going into the 2012 season.
New Bears coach Marc Trestman and his offensive staff made it clear upon arrival that the team will now stress versatility from the tight end position.
"We need a tight end that can threaten the defense. We need a tight end that can create stress in the middle of the field or wherever we place him because we're going to line him up next to the tackle,"
Bears tight ends coach Andy Bischoff said back when the team introduced the new staff. "We're going to line him up outside the numbers. We're going to line up in the backfield, and we're going to expect the defense to figure it out.
"So we need a receiver who can catch the ball, and we need a receiver who can block enough to be lined up in the backfield if we put him there. The days of the tight end that blocked and ran flat routes in this league are coming to an end. The guys that just run 3-yard flat routes, that's not going to do anything for you."
According to ESPN Stats & Information, the Bears targeted a tight end on fewer passes (66) than any team in the NFC, and finished with the fewest receptions (33) in the league from the position. When the tight ends did manage to catch a pass, they averaged 3.4 yards after the catch, which ranked 28th in the NFL.
Perhaps Bennett can remedy the lack of production at the position.