Steven Gerrard is seeking cover for first-choice goalkeeper Emiliano Martinez and has identified Randolph as a target, with the 34-year-old having fallen down the pecking order since Alphonse Areola arrived at West Ham.
Areola, who signed on loan from Paris Saint-Germain, has played in the Europa League and National Cup competitions, keeping six clean sheets, and is seen as Lukasz Fabianski’s long-term successor.
West Ham also has a fourth senior keeper, David Martin, in the books, meaning they would allow Randolph to move on to Villa Park.
Villa is backing Gerrard in the transfer market this month and has already signed former Liverpool playmaker Philippe Coutinho from Barcelona on loan.
The Midlands club are also moving forward with a decision for Everton full-back Lucas Dinge after West Ham deemed his current salary of up to £ 130,000 a week too high.