West Ham, West Brom and Crystal Palace have also been linked with the 29-year-old but Villa manager Paul Lambert has moved to the head of the queue to sign the Englishman.
Richardson has a year left on his current Craven Cottage deal but is desperate to stay in the Premier League following the club's relegation to the Championship last season.
Despite finishing second bottom, the 2013/14 season proved his most productive in a Whites shirt as he scored four goals in 31 league appearances.
Richardson was spotted as a youngster by West Ham but moved to Manchester United in 2002 without playing a first-team game at Upton Park.
He spent five years at Old Trafford before joining Roy Keane's Sunderland in a £5.5m deal. In August 2012 he returned to London to join Fulham for an undisclosed fee.
Richardson, who famously helped West Brom avoid relegation in 2005 - scoring three goals in 12 appearances while on loan at the Hawthorns - has been capped eight times by England.