Keane had been one of the favourites to take over from Neil Lennon at Celtic but he withdrew himself from contention on Monday.
And it now appears he is on his way to Villa Park to assist Lambert, a role that would also allow him to continue working alongside Martin O'Neill with Ireland.
Villa are looking for an assistant after Ian Culverhouse was sacked last month, along with head of football operations Gary Karsa.
No official news on Keane's future is expected until he returns from the United States where the Republic are taking on World Cup qualifiers Costa Rica and Portugal - the last of those games on June 10.
Keane has not managed in England since January 2011 when he was fired by Championship club Ipswich; he resigned from his previous role with Sunderland at the end of 2008.
The 42-year-old returned to the coaching ranks last November when he was unveiled as part of O'Neill's backroom staff after the former Villa boss replaced Giovanni Trapattoni.