The midlands club have discussed the possibility of signing Keane in a move similar to the one which has seen his Galaxy team-mate Landon Donovan rejoin Everton.
Villa boss Alex McLeish has admitted he has limited funds to spend during the January transfer window.
But he is a long-term fan of the Republic of Ireland international and last season came close to signing him when Birmingham manager in a £6 million deal with Tottenham.
The two clubs agreed a fee but personal terms proved a stumbling block and Keane joined West Ham on loan before making the switch to the United States during last summer.
Villa also came close to signing Keane, 31, when he left Wolves 12 years ago but could not agree a fee and he joined Coventry.
If Keane came to Villa, he would link up with Ireland team-mates Shay Given and Richard Dunne.
It is believed Keane wants to stay sharp for the Euro 2012 finals by continuing to play before the Major Soccer League season begins in March.
But terms with Keane and Galaxy have yet to be finalised.
In the immediate future, however, Keane will be training with his old side Tottenham.
The White Hart Lane side released a statement on Tuesday saying: "The club can confirm that Robbie Keane will train with the first-team squad up until January 20 in order to build up his fitness ahead of returning to LA Galaxy for the start of the new MLS season."