Crystal Palace, Everton and Sunderland are also interested in Bahoui but The Saints are willing to spend money after losing key players already this summer.
Rickie Lambert, Luke Shaw and Adam Lallana have all left the club for an estimated £50 million and new Southampton manager Ronald Koeman will look to rebuild the side ahead of the new season.
Deals for Eredivisie duo Dusan Tadic and Graziano Pelle are thought to be close to completion and the addition of Bahoui as well would give the club attacking options.
The Swedish international joined AIK in November 2012 and has started the new season well with four goals and two assists in 11 appearances, including a brace in their last game before the World Cup break.
The direct wide-man can play on either wing or as a striker and at just 23, he has plenty of potential to develop in what will be a new-look Southampton side.