He was with the Green Bay Packers in training camp last year.
Mulligan is primarily a blocking tight end at 6-foot-4, 267 pounds. He has played in 65 games with 24 starts and 16 catches for 160 yards and two touchdowns.
He spent his first NFL season on the practice squad of Miami and Tennessee and then was claimed off waivers in 2009 by the Jets after being released by Tennessee during training camp.
Last season, Mulligan appeared in 15 games with four starts for the Patriots, catching two passes for 16 yards.
Bellamy has appeared in eight NFL games in two seasons with the Chiefs, Chargers and Redskins. The 6-foot, 206-pounder averaged 20.2 yards on five kickoff returns with Kansas City in 2012, but has yet to catch a pass in a regular-season game.
Bellamy, 24, entered the league with the Chiefs in 2012 as an undrafted free agent from Louisville. As a senior in 2011, he caught 24 passes for 280 yards and two touchdowns.
Bellamy is one of eight receivers on the Bears roster, joining Brandon Marshall, Alshon Jeffery, Domenik Hixon, Marquess Wilson, Eric Weems, Chris Williams and Terrence Toliver.
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Aaron Wilson covers the Ravens for The Baltimore Sun.