The 30-year-old England international is certain to leave Upton Park this summer after the club suffered relegation from the Premier League last season.
Parker, who is under contract until 2014, has been linked with a return to former club Chelsea, with West Ham admitting they may agree a loan deal.
But Villa's bid could persuade Hammers co-owners David Gold and David Sullivan that a permanent sale is the better option.
Villa will have money to spend when the £19million sale of England winger Stewart Downing is confirmed and they are understood to have bid for Wigan's Charles N'Zogbia.
New Villa boss Alex McLeish is attempting to make changes to his side and a fee has also been agreed with Manchester City for goalkeeper Shay Given.
Parker, the 2010/11 Football Writers' Association Player of the Season, is also on McLeish's wanted list and it remains to be seen how West Ham respond.
They will need to cut their wage bill after dropping into the Championship and it was reported on Thursday that they wanted Chelsea to pay £4m plus Parker's wages for a loan.
Chelsea are in need of midfield cover after Michael Essien was ruled out for up to six months following knee surgery, but boss Andre Villas-Boas may not want a permanent signing.