The 23-year-old, part of his country's 2010 World Cup squad, spent the second half of last season on loan at Villa from Borussia Monchengladbach.
Villa had an option to buy the player at the end of the campaign under the terms of the initial agreement, but the German club have allowed them until a replacement for Gerard Houllier is in place to make a final decision.
Bradley made just one start - against Manchester City in the FA Cup - and two substitute appearances during his four-month stint at Villa Park.
The ?£6million January signing of Jean Makoun and the re-emergence of skipper Stiliyan Petrov and Nigel Reo-Coker, who has since left on a free transfer, meant he had few opportunities.
Villa are continuing their search for a new manager with former England coach Steve McClaren and ex-Tottenham boss Martin Jol among the front-runners.
Club owner Randy Lerner hopes to make an appointment by next week.
Source: PA
Source: PA