The Brazilian has been heavily linked with a return to the Premier League following a troubled time at the Nou Camp, and United are reported to be leading the race.
The 26-year-old put in one of his best performances in a Barca shirt as he found the net during last night's 5-1 rout against Lyon that put the Catalan giants through to the quarter-finals.
However he did little to commit his future to the club after the game, and seems open to a move away, when the transfer window re-opens in the summer.
"In the last few weeks, months even, I've not been playing well. But I still want to improve," Coutinho revealed after the game.
"I came to Barcelona to give my best. I live for the moment. You never know what the future holds. I am happy with my performance today."
A return to former club Liverpool has been ruled out, so United appear to be in the box seat to do a deal, although Chelsea have also been mentioned.