Jose Mourinho wants to add a centre-back to his squad in the summer and the Mirror are claiming he has sent scouts to watch the Greece international in recent weeks.
The report goes on to claim that United are still interested in Tottenham ace Toby Alderweireld but see Papastathopoulos, 29, as a more likely target.
Alderweireld, 29, is stalling on signing a new contract with Spurs but the club are under no pressure to sell the Belgian international.
Papastathopoulos, now rated at £23.8million, joined Dortmund in 2013 when now-Liverpool boss Jurgen Klopp was in charge.
Since then he has become one of the Bundesliga club's most reliable players and has had interest from a number of clubs.
Chelsea have also reportedly been monitoring Papastathopoulos but the Mirror are adamant United are leading the race.
Arsenal have also been linked with Papastathopoulos who was reported to want to leave Dortmund in January.
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He was said to want to quit the German giants to try and win more trophies.
Papastathopoulos is under contract until 2019 and Dortmund will know they could cash in during the summer.
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