The 28-year-old midfielder was on loan at Villa Park and their boss David O'Leary had been hoping to extend his deal until the end of the season.
However, Leeds will only allow the Norwegian to leave on a permanent deal and O'Leary has indicated they do not have the funds available to sign him.
Bakke returns carrying a slight hamstring injury but will not be able to play for the Whites anyway until his loan registration with Villa officially finishes at the end of January.