The 23-year-old is widely expected to leave Leipzig during this summer's transfer window, and Liverpool are believed to be the favourites to sign the Germany international.
The Reds have allegedly already offered the forward a five-year contract having agreed to pay the €58m (£49.1m) release clause in his current deal.
However, according to The Athletic, United have also made an offer for the German as they attempt to bring a new long-term striker to Old Trafford during this summer's transfer window.
The report claims that Barcelona have also made an attempt to secure a deal, but Werner is holding out for a switch to Liverpool as he wants to work under Jurgen Klopp.
The former Stuttgart youngster has scored 27 times and registered 11 assists in 33 appearances at club level this season, including four goals in seven Champions League outings.