The France international has been linked with a move away from the Nou Camp in recent months.
Umtiti, 24, is under contract until 2021 and talks over a new deal with Barca have stalled.
And according to Spanish news outlet Don Balon, Umtiti is demanding a bumper pay rise to stay at the Nou Camp.
They also claim the centre-back has an offer on the table from Jose Mourinho to join United.
Don Balon add that Barca talisman Messi fears Umtiti could quit La Liga and head to Old Trafford.
Messi is a huge admirer of the defender and he has told the club do give in to his contract demands.
“Lionel Messi is worried that Barcelona star Samuel Umtiti could join Manchester United”
Barca boss Ernesto Valverde recently claimed Umtiti will pen a new long-term deal in La Liga.
“Umtiti is a great player and we expect him to stay at Barca for many more years and for him to enjoy his football with us,” he said.
“To an extent, the buyout clause is made high so that you can't buy the player.