Keane held an informal meeting with Lambert last month with a view to replacing Ian Culverhouse, who was dismissed by the club following an internal investigation.
It is understood Lambert would be happy for Keane to continue in his role as number two to Republic of Ireland boss Martin O'Neill if he were to take the Villa role.
Keane is due to link back up with the Ireland squad in Dublin late on Tuesday before flying out on Wednesday for their two-game tour of the United States.
Earlier this week, Keane indicated he did not wish to be considered as a potential replacement for Neil Lennon at Celtic.
A source close to the club told Press Association Sport Keane had "asked not to be considered for the position".
Lambert is keen to get a permanent assistant on board as he rebuilds following Villa's disappointing 15th-placed finish last season.
The Villa job would offer Keane a route back into domestic management for the first time since he left Ipswich in January 2011.
- Sports & Recreation
- Ian Culverhouse