The fee for the 26-year-old is believed to be more than the £8m the Baggies paid Hull for Chester when he moved to The Hawthorns in July 2015.
Former Manchester United trainee Chester made 13 appearances in the Premier League last season.
He featured in every game as Wales reached the semi-finals of Euro 2016.
Villa, relegated from the top flight last season, equalled the second-highest fee they have ever paid for a player when they signed striker Ross McCormack from Fulham for £12m earlier in August.
But they have made a disappointing start to the season, losing their opening Championship at Sheffield Wednesday and then crashing out of the EFL Cup to League Two Luton.
Rob Gurney, BBC WM
"It's clear from Wednesday's risible performance at Luton that Villa need defensive reinforcements urgently, and Chester would fit the bill.
"He had an accomplished Euro 2016, and after starting only nine Premier League games for the Baggies last season - most of those at left-back - it's clear that Tony Pulis doesn't see him as part of the future at The Hawthorns, whatever that turns out to be under the new Chinese ownership.
"I understand Villa remain in the box seat to sign Chester, with an increase on the £8m Albion paid last summer representing good business for them.
"Now there's a fairly desperate need for recruitment at West Brom to try to appease an increasingly restless fan base."
Premier League side West Bromwich Albion have accepted Championship club Aston Villa's bid for Wales defender James Chester, reports BBC WM.