The midfielder has been frozen out in Paris after refusing to sign a new contract and will leave the club on a free at the end of the season.
Barcelona were thought to be close to signing him but their interest appears to have cooled after they agreed to sign Frankie de Jong from Ajax for £65million when the current campaign ends.
Tottenham, Arsenal and Chelsea are also said to have held talks with the 23-year-old.
And The Sun say United have now entered the race because manager Ole Gunnar Solskjaer is concerned by his ageing midfield.
Nemanja Matic will be 31 by the time next season starts while Juan Mata turns 31 next month.
The report quotes a United source as saying: “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.”
While Solskjaer is technically still interim manager, the decision to give the Norwegian the job full-time is understood to have already been made.
It is claimed he is drawing up an extensive list of targets for an assault on the Premier League title next term.
Rabiot might appeal as he will not cost anything in terms of a transfer fee, although he will reportedly demand a massive signing-on fee of around £10million.
United have the money and are said to be happy to pay it to ward off any other potential suitors for him.
The six-cap France international has not played any football since the beginning of December after being ‘indefinitely benched’ for holding out own a new deal.
Rabiot has stood from and his happy to bide his time and wait for a move.