Following his departure from relegated Leicester City at the end of last season, Evans returned to Man United on a short-term deal to provide defensive cover and maintain his match fitness.
Evans boarded the plane with Man United for their pre-season tour of the United States, featuring in a pair of friendlies against Lyon and Wrexham but failing to earn any minutes against Arsenal or Real Madrid.
The 35-year-old was originally only set to stay on until the end of July, but Erik ten Hag is understood to have been left hugely impressed with Evans, leading to speculation of a potentially longer stay.
According to The Sun, Man United are now weighing up whether to offer Evans a one-year contract for the 2023-24 campaign, given his exceptional fitness levels and leadership qualities.
In addition, Ten Hag is also believed to admire Evans's ability to play out from the back - a necessity for defenders in the Dutchman's system - and further exits could see the Northern Ireland international make a permanent return to Old Trafford.
The Red Devils have already cut Phil Jones and Axel Tuanzebe from their ranks this summer, and the duo of Harry Maguire and Eric Bailly could also depart for the right price in the coming weeks.
Man United are reported to have turned down one West Ham United bid in the region of £20m for Maguire, who has now been stripped of the captaincy after losing his place in Ten Hag's starting lineup last term.
Continued lack of minutes is likely to threaten Maguire's place in any England Euro 2024 squad, but the 30-year-old is supposedly prepared to stay and fight for his place in the team, while a recent wage increase and £50m asking price could also deter suitors.
On the other hand, Bailly - who spent the 2022-23 season on loan at Marseille - is widely expected to bid farewell to the Theatre of Dreams, and the Ivorian is rumoured to be closing in on a move to Saudi Pro League side Al-Nassr.
Victor Lindelof's next career move has also been the subject of much debate, but the Swede found himself above Maguire in the defensive order last term and may stick around for a little while longer.
Evans previously turned out for Man United for 11 years between 2004 and 2015, during which time he won three Premier League titles, in addition to the Champions League, Club World Cup, EFL Cup and Community Shield.
The Northern Irishman produced seven goals and seven assists in 198 contests for the Red Devils during his first spell at the club, which came to an end with a move to West Bromwich Albion eight years ago.
Evans won another FA Cup and Community Shield during his five-year stint with Leicester, but he has struggled with injuries over the past two years and only started 12 Premier League games last term.
Man United have already added Mason Mount and Andre Onana to their books this summer, and the Red Devils are now reported to have agreed a £64.4m deal - potentially rising to £73m - for Atalanta BC striker Rasmus Hojlund.