The 23-year-old midfielder is free to leave the Ligue 1 giants this summer when his contract expires.
The France international is refusing to sign a new deal and has held talks with Barcelona, Tottenham, Chelsea and Arsenal.
And the Red Devils are now showing interest in him, despite a number of off-the-field issues — including the player going out partying after their Champions League defeat to United.
Ole Gunnar Solskjaer is preparing a list of targets for when the transfer window opens in the summer.
He has earmarked Rabiot as a potential replacement for his ageing midfield with Nemanja Matic, Juan Mata and Ander Herrera all getting on.
It is understood he would be looking for wages of around £200,000-a-week to sign with the Red Devils.
He would also be looking for a staggering £10m signing-on fee, which United would be willing to pay.
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An Old Trafford source said: “There is interest there.
“His advisers look to be offering him around though.
“The boss is well aware of his ability.
“We are on the lookout for a midfield link player for the long-term.”
Manchester City are looking at Rabiot with concerns over Ilkay Gundogan signing a new contract.
He had a brief spell in the Etihad youth team but returned to France because of homesickness.
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