The Spanish publication claims that Jose Mourinho tried to lure the France international to Old Trafford ahead of the second half of the season by offering him a stunning deal worth around €6m (£5.35m, $7.4m) per year after taxes – equivalent to £103,000-a-week.
Sport says that United were also ready to meet his €60m (£53.4m, $74m) release clause to complete the transfer after Manchester City and Liverpool paid in excess of that to bolster their back-lines with the respective €65m and €80m [BBC Sport] additions of Aymeric Laporte [Sky Sports] and Virgil van Dijk.
The reports claims that Barcelona officials reacted quickly and promised Umtiti a new deal to convince him to turn down the overtures from Mourinho and stay at the Nou Camp.
The 24-year-old defender has developed into one the best centre-backs around Europe since his arrival from Lyon in the summer of 2016, forming an impressive partnership in the heart of the back-line alongside Gerard Pique and being a key player in the France national team.
Sport says that Barcelona have already made contact with Umtiti's agent in a bid to secure the long-term future of the defender until 2023 or 2024.
Captain Andres Iniesta, Lionel Messi, Gerard Pique and Sergi Roberto have already inked new contracts in recent months and the idea is for Umtiti to put pen to paper on the deal ahead of next season.
Sport claims that they want to complete the agreement as soon as possible amid fears that more clubs could try to sign him in the summer if he enjoys a good World Cup with France.
That's why Barcelona have already opened talks with Umtiti's agent but the Spanish publication suggests that the €6m-a-year offer made by United is going to force the Catalans to pay him much more than they originally planned.
Nevertheless, Sport points out that Umtiti is currently earning much less at the Nou Camp and if Barcelona agree to match the proposal made by United he would become into one of the best paid players in Ernesto Valverde's squad.
The report adds that the La Liga giants believe there is still "a long way to go" before fans will see the Frenchman putting pen-to-paper on the new contract.