The Ivory Coast international right-back has been linked with a move to Old Trafford this summer.
He faces an uncertain future at PSG following the arrival of Dani Alves from Juventus.
And, according to the Daily Mail, Aurier is desperate to quit the French giants.
Jose Mourinho is reportedly on the hunt for a right-back this summer.
Monaco utility man Fabinho - who can play there or in midfield - is said to be a target.
But The Independent yesterday claimed that he is proving tricky to sign so Aurier is emerging as an alternative target.
The Daily Mail report that Manchester City and Juventus have also been offered the chance to sign Aurier.
Aurier is actively seeking an exit from PSG and is not with the club on their pre-season tour of the United States of America.
PSG boss Unai Emery confirmed he had been left out of their International Champions Cup squad.
"I spoke with Serge to tell him that I wanted him to stay," Emery said.
"He told me that he wants to leave and he preferred staying in Paris to prepare for his departure."
Aurier, under contract until 2019, has made 81 appearances for PSG.
