Weghorst is on loan at Besiktas from Burnley, and the Turkish side are keen to keep the striker until the end of the season.
Sources have told ESPN that United want to negotiate their own loan deal but first Besiktas would have to agree to send Weghorst back to Turf Moor.
United have money available for a small loan fee following Cristiano Ronaldo's departure during the World Cup, but there are significant budget limitations after spending more than expected on transfers last summer.
Sources added United are in talks with Burnley about a deal which would see Besiktas receive a payment to allow Weghorst to leave.
Besiktas said in a statement on Monday: "The claim that there is an exit clause in our football player's contract, stating that his contract can be terminated with a compensation of €2.5 million in case of an offer from the Premier League, is fictitious.
"However, the news that the player in question will leave the team on Tuesday is completely untrue. The initiative regarding Wout Weghorst belongs entirely to Besiktas."
Sources have told ESPN that Weghorst is viewed by Erik ten Hag as a suitable short-term option to compensate for the loss of Ronaldo.
Ten Hag is keen to add a physical striker to his squad to offer a different option to players he already has and 6ft 6in Weghorst, who has scored eight goals in 16 league games for Besiktas this season, fits the bill.
Weghorst wants to move to Old Trafford, sources have told ESPN, but any deal depends on Besiktas' willingness to part company with the Dutchman midway through the season.
United have other options to strengthen Ten Hag's squad in January but are focused on signing a forward only until the end of the season to allow the club to pursue more long-term targets ahead of the summer transfer window.