The 24-year old Japan international has not lived up to the £24million fee Sir Alex Ferguson paid for the midfielder in 2012.
But Juan Mata’s £37.1million arrival from Chelsea in January means Kagawa has been pushed further down the Old Trafford pecking order, with Besiktas reportedly willing to end his Premier League nightmare.
Borussia Dortmund are also interested in brining Kagawa back to the Bundesliga, a move that is thought to be preferable to the Japanese midfielder.
David Moyes is hoping to recoup as much of the original fee paid for him as possible but may propose a deal that would see Ilkay Gundogan go the other way.
But if Dortmund are unwilling to part ways with Gundogan then United will sell to the highest bidder, with Besiktas the currently the favourites for his signature.
Meanwhile, Moyes has been dealt a huge blow in his pursuit of Bayern Munich’s Toni Kroos.
Bayern boss Pep Guardiola has revealed Kroos will be extending his current contract with the Bundesliga club, all but ending United’s hopes of bringing him to Old Trafford in the summer.
He said: “We all want him to prolong, and he also wants that, I am very positive about it.”