Striker Bothroyd, capped by England last November, has regularly been linked with a move to the Premier League and the Bluebirds' leading scorer looks set to get his wish.
Winger Burke will also leave the Welsh club as a free agent along with Adam Matthews, who will join Celtic, Gavin Rae and Martin John.
Young duo Jonathan Meades and Jordan Santiago have been offered professional contracts, while Academy trio Ibrahim Farah, Nathaniel Jarvis and Alex Evans have also been offered deals.
Furthermore, Sameth Al-Basith and Steffan Jones have been released.
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