Sanchez has struggled at Manchester United since joining last January from Arsenal despite his incredible £500,000 a week wages.
And United are already believed to be considering letting him go if they can get a buyer.
The Sun report that French giants PSG are keen and could make a move in the summer.
However it is claimed they want a bargain deal for the Chile winger, and could even ask for him on a free.
A source told the Sun: “There has been some whispers that PSG want to take Alexis off of the club next summer.
“It’s no secret he has struggled and not lived up to the expectations. He isn’t getting any younger and it might never work for him.
“There was interest before the club signed him but he preferred staying in England at the time. That may have changed now.
“One of their top stars could go and it’s not beyond the realms of possibility that a deal could be struck between the two clubs.”
Sanchez has even dropped out of Jose Mourinho’s starting Xi at United after a series of underwhelming displays this term.
The 29-year-old was left out of the squad to face West Ham, came off the bench in the dramatic win over Newcastle and the draw against Chelsea but was an unused sub against Everton on Sunday.
The 32-year-old watched on as goals from Paul Pogba and Anthony Martial secured the points for United as their gradual return to form continued.
United picked up their second win in three games since the defeat to West Ham in the league but remain down in eighth position in the table.
They face sixth place Bournemouth this weekend in another huge test for Mourinho.
The pressure has eased on Mourinho in recent weeks and he seems to be safe in his job for now.
But with difficult games against Juventus and Manchester City to come the pressure could be back on soon.