Malacia did not make a single appearance for the Red Devils during the 2023-24 campaign, as he required two separate operations on a knee problem which has caused concern at Old Trafford.
The Netherlands international was not in the batch of images that Man United released of their players last week, fuelling claims that he was still some way from returning to action.
However, on Sunday morning, Man United revealed that the left-back was able to train individually last week, with the 24-year-old working "under the guidance of medical staff".
"There was encouraging news from Carrington on Friday after Manchester United defender Tyrell Malacia progressed his fitness programme," read a statement on the club's official website.
Malacia trained individually at Carrington last week
"Young left-back Malacia returned to base earlier this week, after working hard throughout his summer break. Tyrell has this week been pictured training individually on the grass under the guidance of medical staff.
"The Netherlands international joined United from Feyenoord in 2022 and has made 39 appearances so far. He is doing everything he can to regain full fitness."
Malacia will not be available for Man United's first pre-season friendly against Rosenborg on Monday, and it is difficult to imagine him featuring before the end of the month, but there is hope that the Dutchman will be available towards the start of the 2024-25 campaign.
The defender had a successful first season at Man United after arriving from Feyenoord in the summer of 2022, making 39 appearances in all competitions, including 22 outings in the Premier League.
Malacia had initially been expected to return at the start of this year, but various setbacks meant that he was not able to feature in the second half of the campaign.
Will Man United sign a left-back this summer?
Luke Shaw also had his injury problems last season, only making 15 appearances in all competitions, but he has returned to fitness for England at Euro 2024.
The 29-year-old will be Man United's first-choice left-back next season providing that he is fit and available, but Man United could still potentially sign a new player in that area of the field this summer.
Bournemouth's Milos Kerkez has admirers at Old Trafford, while the club are believed to be considering making a move for Feyenoord's David Hancko.
There have also been suggestions that Tottenham Hotspur's Sergio Reguilon, who spent the first half of last season on loan at Man United, could return to Erik ten Hag's side ahead of the new season.
However, the club are currently focusing on the middle of their defence, having secured a deal for Bologna's Joshua Zirkzee, with an announcement on the striker expected to arrive in the near future.