Bannan broke into the Villa team earlier in the campaign but has slipped out of the reckoning in recent weeks.
The 21-year-old midfielder has been told he is part of Villa's future and signed a new contract recently but he could be allowed to go to Elland Road to gain valuable experience.
High hopes: Leeds want Bannan (right) to boost their promotion chances
Villa boss Gerard Houllier will be more likely to let Bannan leave if his side's FA Cup run ends against Manchester City on Wednesday night.
The youngster made his Scotland debut earlier in the season against the Faroe Islands and won his second cap against Northern Ireland last month.
Downing stakes his claim for an England place in front of CapelloWatch all the Premier League goals every week on our brilliant video playerSign me up! Pires wants to stay at Aston Villa for another seasonAll the latest Aston Villa news, features and opinion
Explore more:People: Simon Grayson, Gerard Houllier Places: Leeds, Scotland, Northern Ireland