Sources have told ESPN that United have held discussions with Burnley and Weghorst is keen on the transfer if terms can be agreed.
The 30-year-old scored two Premier League goals for Burnley last season but has scored eight times in 17 appearances for Besiktas this term.
Weghorst appeared to wave goodbye to Besiktas supporters after scoring in a 2-1 win over Kasimpasa on Saturday.
United are looking to sign a striker in the January transfer window following the departure of Cristiano Ronaldo.
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The club are only looking at low cost deals after a larger than expected transfer spend last summer and have considered targets including Memphis Depay, Alvaro Morata, Joao Felix and Vincent Aboubakar.
Sources have told ESPN that United have also been restricted in terms of targets due the club only wanting to make a short-term signing until the end of the season in order to free up resources and squad space for the signing of a long-term solution to the team's goal-scoring problems in the summer.
A move for AC Milan's Olivier Giroud has been considered by United due to the former Chelsea and Arsenal striker having just six months to run on his contract at San Siro, but sources have said that the prospect of the France international being hit by a sizeable tax bill by leaving Italy before the end of his contract ruled out the prospect of a deal for the 36-year-old.
Sources also said that the market is currently stocked with players deemed to be 'cast-offs' or those with questionable fitness records, but Weghorst is regarded as being a player who can solve a short-term problem and is one who is familiar to Ten Hag following his time in Eredivisie prior to his move to German team Wolfsburg in 2018.