The Manchester United legend is currently second in command to Martin O'Neill at ROI, but was heavily linked with the vacant Celtic job following Neil Lennon's departure.
However the Irishman declined the Scottish champions' overtures to stay with the side he once captained, but is now attracting the attention of Villans boss Lambert.
According to The Mirror, Lambert is keen to add Keane to his backroom staff following the sackings of Ian Culverhouse and Gary Karsa last month.
It is believed that Keane is tempted by the offer, and it would enable him to maintain his current roles as the Irish assistant, and as a pundit for ITV Sport.
The former Republic of Ireland and United skipper is seen as someone who would command instant respect of the Villa dressing room.
Lambert is understood to be resigned to another summer of bargain hunting following owner Randy Lerner's decision to put the club up for sale.
The American is reportedly struggling to offload the club and has slashed the asking price by a whopping £50 million as he looks to negotiate a quick sale.
The Mirror reports that Lerner was hoping to open the bidding at around £200 million, but will now settle for £150 million after slashing the fee by a quarter.