The Red Devils are yet to add to their defensive ranks this summer, but Phil Jones and Axel Tuanzebe were allowed to depart upon the expirations of their contracts.
The out-of-favour Eric Bailly is also believed to have been the subject of two bids from Saudi Arabian clubs, while Harry Maguire's future remains a major talking point.
West Ham United have supposedly failed with one £20m bid for the England international, who was stripped of the captaincy by Erik ten Hag earlier in the summer.
While Man United are not planning to force Maguire out of the club, he could be allowed to depart for the right price amid concerns that a lack of game time could hinder his chances of a Euro 2024 call-up.
Maguire's possible exit would leave Raphael Varane, Lisandro Martinez, Victor Lindelof and Luke Shaw - a left-back by trade - as Ten Hag's central options, while it remains to be seen if Jonny Evans will be kept on for the 2023-24 campaign.
Having now finalised the signing of Rasmus Hojlund from Atalanta BC, Man United could turn their attention to defensive reinforcements, and Nacional - via The Mirror - claims that Christensen has emerged as a target.
The Denmark international left Chelsea as a free agent last year to join the Catalan giants, and he won the La Liga title and Supercopa de Espana during his debut campaign.
Amid a couple of injury concerns, Christensen scored one goal and set up one more in 32 matches for La Blaugrana last term, starting 22 top-flight games and the second leg of his side's Europa League playoff round defeat to Man United.
The 27-year-old is under contract at Camp Nou for another three seasons, but Xavi has added Inigo Martinez to his ranks to compete with Christensen, Ronald Araujo, Jules Kounde and Eric Garcia for central starts next season.
The report claims that Man United have made an official bid worth €50m (£43.1m), which would see the Red Devils pay €35m (£30.2m) up front in addition to €15m (£12.9m) in bonuses.
Barcelona are supposedly happy to sanction Christensen's exit for around the €50m (£43.1m) mark, but Man United will need to commit to paying more up front if they are to get their man.
During his time in the Premier League with Chelsea, Christensen made 93 top-flight appearances for the Blues and won the Champions League under Thomas Tuchel before departing.
Real Madrid's David Alaba and Everton's Jarrad Branthwaite have also been linked with moves to Old Trafford this summer, but it would supposedly take a £25m fee to sign the latter.
Axel Disasi was on the Red Devils' radar too, but Ten Hag's side lost out to Chelsea in the race to sign the former Monaco and Reims man.