Paul Pogba's agent has offered the France international to Liverpool, according to reports.
The midfielder is under contract at Manchester United until 2022 and has hitherto shown little interest in signing an extension.
United may therefore opt to sell Pogba this summer, rather than risk losing him for nothing in 12 months' time.
PSG are said to be monitoring the situation and could submit a formal offer in the coming weeks.
The Reds are said to be on the lookout for a new midfielder, having lost Georginio Wijnaldum to PSG.
Jurgen Klopp's side are keeping tabs on Eduardo Camavinga and Houssem Aouar, the report states.
And Mino Raiola, the Italian superagent who represents Pogba, has now offered his client to United's arch-rivals.
It is almost impossible to envisage the Frenchman moving to Anfield this summer, however.
The last player to move directly between United and Liverpool was Phil Chisnall in 1964.
The Red Devils would probably be unwilling to sell Pogba to another Premier League, and certainly not to their biggest adversaries.
Having said that, it is not completely out of the question that Pogba might one day be wearing the red of Liverpool rather than that of United.
His current employers will have no control over Pogba's next destination if he departs on a free transfer next summer.
Of course, that does not necessarily mean that a move to Liverpool is realistic - but United would no longer be able to stop it.
PSG still looks like the most likely destination, but the French giants might prefer to wait until 2022.
They have already brought in Wijnaldum, Sergio Ramos and Gigio Donnarumma on free transfers this summer, and may be hoping to repeat the trick with Pogba next year.
Raiola's mischief-making is unlikely to mount to anything on this occasion.
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