Dembele is out of contract next summer and would be available as a free agent as thing stand.
According to Mail Online, United have touched base with the 24-year-old's agent.
Dembele joined Barca from Borussia Dortmund for €105 million (£97 million) in 2017, making him one of the most expensive players of all time.
However, he hasn't enjoyed the greatest of spells, with his time at Camp Nou disrupted by injuries.
Barca were believed to be open to selling Dembele this summer, amid their severe financial problems, but no move materialised.
However, that may have been down to the fact the France international has been out since June after surgery on a knee injury he sustained at Euro 2020. He's set to remain on the sidelines for at least another month yet.
Would a move to Old Trafford in nine months' time or so allow Dembele to truly live up to the serious promise he showed as a youngster, though?
Well, even if he weren't so injury-prone, United are already blessed with talent in the wide areas.
Mason Greenwood - who, even at the age of 19, looks like he could become a superstar - may eventually become a centre-forward, but Cristiano Ronaldo's return means he's going to have to continue on the right - Dembele's usual position - in the short to medium term.
And Jadon Sancho - who's bound to get up and running for his new team sooner rather than later - is arguably just as much of a threat from the right as he is from the left.
Even if he's available on a free transfer, Dembele will likely command a hefty salary, and attack just isn't a priority for United at the moment.
Subscribe to FourFourTwo today and get three issues delivered for just £3. The offer ends October 17, 2021.
GOLDEN BOOT RACE Who are the top scorers in the Premier League this season?
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.