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The 24-year-old is reportedly considering his future at Old Trafford after starting this month's 4-1 defeat to Manchester City on the bench.
According to the Manchester Evening News, sources close to Rashford have claimed his phone 'hasn't stopped ringing' since the rumours emerged.
It's believed Rashford doesn't want to leave his boyhood club and it could be a way to secure a new long-term contract.
The England international's deal runs until 2023, but United have the option to extend it by an extra year.
Rashford has featured 26 times in all competitions this season but only managed five goals.
Barcelona would like to add Rashford to their rejuvenated front line and make him a key player.
The Catalan giants signed Ferran Torres and Pierre-Emerick Aubameyang on permanent transfers in January while Adama Traore is on loan from Wolverhampton Wanderers.
Dembele is out of contract this summer and Luuk de Jong will be returning to Sevilla.
There are question marks over the future of Memphis Depay too, despite the former United man only arriving last summer.
Rashford also has admirers at Liverpool but it's unlikely he'd be allowed to join United's bitter rivals.
No player has moved between United and Liverpool since 1964.
Anthony Elanga's emergence and Jadon Sancho's summer arrival has seen Rashford drop to the bench in recent weeks.
Elanga and Sancho started Tuesday's Champions League elimination at the hands of Atletico Madrid, with Rashford coming on in the second half.
Interim manager Ralf Rangnick said last week he's happy for Rashford to leave if he's unhappy, but had no indication he was.
"I don’t think this is an issue and he never said anything about that since we spoke yesterday [last Thursday]," Rangnick said per the Guardian.
"But the [transfer] window is closed now. Even suppose he and his agents or his family at one stage of the summer think about maybe going to another club, well then he can do that but in the summer. Now the window is closed.
"I spoke with him yesterday and the day before so I don’t know anything about him being unhappy. He didn’t say anything to me, he said to me that he is still happy and that he wants to perform on a high level for the club.
"There was no mention of him not being happy here."