He joined the club on a two-year deal under the Bosman ruling, having left Cardiff City at the end of the 2010/11 campaign.
The Glaswegian came through the ranks at Rangers and gained UEFA Champions League experience, as well as an SPL title and Scottish League Cup, during a nine-year stay at his boyhood club.
Burke headed to South Wales in January 2009 and quickly became an integral part of a Cardiff side that came close to achieving promotion to the top flight on several occasions.
He had a tremendous first season at Blues, where he made 61 appearances, scored 14 goals and provided 18 assists to the cause. His efforts saw him sweep the board at at the 2011/12 Player Awards Dinner, deservedly claiming the Players' Player of the Season, Player of the Season Award and Junior Blues' Player of the Season Award.
The club took up a one-year option on Burke in the summer of 2013, that runs until June 30, 2014.