The Daily Star Sunday says United have already had two bids of £26m and £30m rejected.
But vice chairman Ed Woodward is ready to go back in with a third offer of £40m.
Barca boss Gerardo Martino said: “I would prefer him to stay with us and I dare say he will but it’s his decision.
“He is one of the best players in the squad. I think he will stay. But there’s a part that he has to decide.
“Cesc will decide the situation – but it has not occurred to me that I won’t be able to count on him.”