The 32-year-old has one year left on the two-year deal he signed after joining the Baggies last summer following his release by Manchester City.
Villa have been active in the transfer market since offloading star striker Christian Benteke to Liverpool for £32.5m.
Tim Sherwood is keen to bolster his squad and, after completing the signing of forward Jordan Ayew from Lorient, he is closing on deals for Emmanuel Adebayor from Tottenham, Rudy Gestede from Blackburn and Nantes midfielder Jordan Veretout, with the trio soon to undergo medicals at Villa Park.
Veteran Dutch defender Ron Vlaar’s contract with Villa has expired and Sherwood is keen to secure an experienced replacement.
Former England international Lescott made 29 appearances in his debut season at the Hawthorns but may struggle to match that total this season following the signing of James Chester from Hull City for £8m.