The report claims that the German giants believe Barcelona will sell Turan for the right price and that a deal may be possible.
Dortmund lost Mkhitaryan to United but the Red Devils are also keeping tabs on Turan according to Fanatik and that he is available for £26 million.
United boss Jose Mourinho was reported to have wanted Turan at Chelsea last season before his move to Barcelona.
Turan has also been linked with Arsenal and Chelsea over the past few months but Barcelona club president Josep Maria Bartomeu recently hinted that Turan would be staying in his most recent press conference.
Bartomeu also denied rumours that the board was unhappy with his performance last season.
The club president said: “The coaching staff count on Arda Turan and we thank both him and Aleix [Vidal] for the patience they showed to go six months without being able to play. As for the sporting section, I’d give them a nine out of 10 because four trophies in one season isn’t bad at all”.
Turan did of course miss the first six-months of the season due to the FIFA transfer ban imposed on the club but made 25 appearances after making his debut in January.
The Turkish international star does have another four-years remanning on his contract and did recently reveal that he was happy to stay and fight for his place in the team.