Umtiti was chased by the Red Devils in the January transfer window but, whilst their pursuit was ultimately unsuccessful, they have not been put off in trying to land the France worldwide.
Jose Mourinho has already confirmed that the club will not be looking to sign an more attacking players at the season's end, suggesting he will be prioritising additions to his midfield and defence ahead of the 2018/19 campaign.
United are willing to meet Umtiti's buyout clause and have offered a contract which is worth double what he is now earning with Barca.
The newspaper says Blues officials have told Enrique they are willing to do all they can to sign the French worldwide as they see him as the flawless long-term replacement for David Luiz.
Enrique has been touted as a potential replacement in the media and the Express are citing reports from Spain that claim the former Barca boss has told Chelsea chiefs that he wants to sign Umtiti if he takes over this summer.More news: Trump holds meeting on video game violence
More news: Marathon Petroleum Corp (NYSE:MPC) Shares Sold by Wells Fargo & Company MN
More news: Rapper Juelz Santana flees from Newark Airport when TSA finds gun
Sport say that the right-hand man of United executive vice-chairman Ed Woodward was in attendance at the Nou Camp to watch Umtiti in action during Barcelona's 1-0 victory over Atletico Madrid on Sunday.
He now has a release clause of €60m (£53m).
The 24-year-old has developed into a key part of the Barca team after joining them from Lyon back in 2016.
Sport also reports that Umtiti is "letting himself be loved" by Mourinho, who has ordered his French national team-mates Paul Pogba and Anthony Martial with charming him to Manchester.