The France international remains under contract at Camp Nou until 2021, but has seen his future called into question as his current deal includes a €60 million release clause.
Several clubs would be prepared to meet that figure to land a talented 24-year-old, with United among those also said to be willing to offer Umtiti a sizeable pay rise if he were to leave Camp Nou.
Barca remain determined to ensure that does not happen, with fresh terms set to be discussed, but club president Josep Maria Bartomeu has left the door open by reiterating that no player is bigger than the team.
He told Cadena Ser: "Umtiti is very important, he knows that, the fans know that.
"He's only in the second year of his contract but our desire is for him to renew his deal.
"But, as directors, we have to make sure [players' wages] are within the salary structure to remain a competitive and sustainable club."
Umtiti joined Barcelona in a €25m switch from Lyon in the summer of 2016.
He has made 81 appearances for the club so far, with just one La Liga defeat suffered during his outings in that competition.
It is easy to see why United would want to bring him onto their books, but he is not the only centre-half being considered by Jose Mourinho.
The Red Devils are expected to bolster their defensive ranks this summer, with the likes of Jonny Evans and Raphael Varane also reported to be on their radar.