The 23-year-old is on the newly-promoted club’s list of defensive targets and it is understood they are now ready to firm up their interest with a formal offer of around £8million.
Targett would provide competition for Neil Taylor, who is currently the only senior left-back in Villa’s squad.
Kortney Hause, who has joined in a £3million deal from Wolves, can play in the position but is more predominantly seen as a centre-back by boss Dean Smith.
Targett made 16 Premier League appearances for the Saints last season.
He had previously been on loan at Fulham during the 2017/18 campaign, where he was part of the team which beat Villa to win promotion in the Championship play-off final at Wembley.
Villa, under Smith, went one better 12 months later, beating Derby last month to secure a top flight return.