The Netherlands international has been cut from the 25-man squad for the upcoming Champions League campaign -- despite the inclusion of injured pair Luke Shaw and Tyrell Malacia -- and risks being omitted from the 25-man Premier League squad when it's announced on Sept. 13.
The majority of transfer windows across Europe are now closed but clubs in Turkey can sign players until Sept. 15 giving United a week to agree a deal.
On Wednesday, Besiktas confirmed the arrival of Eric Bailly on a one-year contract after United agreed to allow the Ivory Coast defender, who had a year left on his contract, to move on a free transfer.
Sources have told ESPN that United are willing to consider any offer for Van de Beek.
There was hope earlier in the summer that the 26-year-old's departure might generate a fee but there's now an acceptance it might have to leave on loan.
Van de Beek, who was nominated for the Ballon d'Or in 2019, still has two years left to run on the five-year deal he signed when he arrived in a £35 million ($43m) deal from Ajax in 2020.
He has endured a difficult spell at Old Trafford, making just 60 appearances in all competitions in three years. He spent the second half of the 2021-22 campaign on loan at Everton but only featured seven times.
He made 10 appearances under Ten Hag -- his former manager at Ajax -- last season but started only two games in the Premier League.
He suffered a knee injury in January which ruled him out for the remainder of the campaign.