The Northern Ireland international is available on a free transfer this summer following his release from Leicester City at the end of June.
Evans started training with the Red Devils in July and was then handed a short-term deal to allow him to feature in their friendlies against Lyon and Wrexham.
The 35-year-old was kept on for the team's final match of their pre-season tour of the United States, and he played the final 30 minutes of the 3-2 loss to Borussia Dortmund on Sunday night.
There has been a host of speculation surrounding the centre-back's future over the last week, as he has allegedly made a big impression on head coach Erik ten Hag and the players due to his leadership skills.
According to The Belfast Telegraph, a decision has now been made to hand the defender a one-year contract at the club that he previously represented between 2004 and 2015.
The report claims that Ten Hag is wary of leaving himself short at the back, as Axel Tuanzebe and Phil Jones have both moved on this summer, while Harry Maguire and Eric Bailly are also being heavily linked with exits.
Both Raphael Varane and Lisandro Martinez have had their injury problems, meanwhile, so Evans will allegedly be brought in as cover for the 2023-24 campaign.
The defender's international teammate Steven Davis has said: "Having played with him for so many years I know the quality he possesses and what he brings with his experience to a dressing room.
"I'm sure that was noticed quite early on having been away from the club for a few years. I wouldn't put it past him extending his stay there beyond pre-season.
"I think we were all surprised when Man United let him go because whenever you play with him and see the quality he has first hand, it is a player every one rates very highly."
Evans made 198 appearances for the 20-time English champions during his first spell at the club, winning the Champions League, three Premier League titles, two EFL Cups, four Community Shields and the Club World Cup.
Man United are currently preparing for their final two friendlies of the summer against Lens and Athletic Bilbao, which take place on Saturday and Sunday respectively, while their 2023-24 Premier League campaign will begin against Wolverhampton Wanderers on August 14.