The 36-year-old Morrison is a veteran of 15 NHL seasons, having played with New Jersey, Vancouver, Anaheim, Dallas and Washington before signing with the Flames in the 2010-11 season.
Morrison has four goals and seven assists in 28 games this season for Calgary and 601 points (200 goals, 401 assists) in his career.
He's reached the playoffs nine times during that span and has eight goals and 21 assists in 58 career playoff games.
“As we projected our line-up going forward we felt it was going to be difficult for us to find a spot for Brendan or get him significant ice-time,” said Flames GM Jay Feaster in a statement. “Given his pending UFA status and the fact that we were not going to re-sign him, we felt it important to try to move him for another asset."
Connelly has 21 goals and 105 assists over four seasons with the American Hockey League’s Rockford IceHogs. He originally signed with the Blackhawks on June 8, 2009.