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The 30-year-old appeared to wave goodbye to fans of the Turkish club after scoring the second goal in a 2-1 win over Kasimpasa in the Super Lig.
Reports claim United wants to bring in Weghorst, who is on loan to Besiktas from Burnley, on a temporary basis until the end of the season.
United manager Erik ten Hag has signalled his need for a striker, with Weghorst a surprising choice.
Besiktas manager Senol Gunes said after the Kasimpasa game: "I have not seen Wout Weghorst's move and I do not know.
"I cannot speak about what I do not know.
"Wout plays from the heart. He contributes. I do not find it appropriate to talk about someone's future, there is no such thing on our agenda."
Besiktas would be reluctant to see Weghorst go, believing it struck a deal with Burnley that would keep him at the Istanbul club for the duration of the season.
"Weghorst is an important player for us. It's not enough to say goodbye to the crowd to say goodbye. He has to talk to the club as well," Gunes said.
The Turkish side would be seeking compensation for any transfer, according to sporting director Ceyhun Kazanci.
He told broadcaster NTV: "I don't understand why Weghorst said goodbye to the fans, why he made such a move.
"He is our football player now. There is no clause in our contract with Burnley that includes returning or leaving at half-time. Our contract ends at the end of the season."
Kazanci said Burnley would need to sound out Besiktas about any change to the loan agreement.
"If we accept, the contract will be terminated," he said. "Otherwise, he won't be able to go anywhere until the end of the season."
Weghorst struggled to make an impact in the Premier League with Burnley in a short spell last season, after a previous prolific time at Wolfsburg in the Bundesliga.
The tall target man came to global prominence at the World Cup, scoring twice as a substitute in the quarter-final against Argentina before the Netherlands was ousted on penalties.