Mourinho has not tried to hide his discontent at United's lack of transfer activity this summer, with Brazilian global Fred the only high profile arrival at Old Trafford so far this summer.
Jose Mourinho's side have been linked with moves for the likes of Harry Maguire as they look to strengthen their backline and Sportsmail reported last week that the 29-year-old Boateng had been offered to United after a move to Paris Saint-Germain fell through.
But, speaking before Sunday's friendly clash with Bayern Munich, Mourinho admitted to MUTV that he fears falling behind the rest of the top six if he is ultimately unable to add to his squad. "On Friday we go with the best possible team".
However, Arsenal are the latest side to be named as a potential destination for Boateng, who could be set to leave Bayern this summer.
"Finally, such a hard pre-season is over".
United were beaten 1-0 by Bayern Munich at the Allianz Arena, five days before their Premier League campaign begins at home to Leicester.
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In his post-match press conference, Mourinho warned the upcoming season will be hard unless they make some more signings before Thursday's transfer deadline.
German newspaper BILD report United are looking to swap Martial for Bayern centre-back Jerome Boateng. They could not have done better than what they did. Fantastic experience for the boys.
"At the same time is was [Robert] Lewandowski on Axel, [Arjen] Robben on Mitchell, [Leon] Goretzka on Garner and [Niklas] Sule on Chong".
Mourinho remains open to selling Rashford, who cost United an initial fee of £36m in 2015, but club chief Ed Woodward retains hope that he can revive his career at Old Trafford.
"Yeah, he wanted it". "Of course, he respected (Wayne) Rooney". Lindelof the same, Jones the same and let's see if one of the others is ready to give us the help - 20 or 25 minutes.