The 21-year-old is expected to leave Old Trafford before the September 1 deadline, with the Red Devils willing to entertain offers for the England Under-21 international.
A recent report claims that Erik ten Hag has given United the green light to sell Garner, who is valued between £15m and £20m.
Garner enjoyed a successful 2021-22 campaign on loan with Nottingham Forest, where he scored four goals and registered six assists in 49 games across all tournaments.
The midfielder was an important member of Steve Cooper's squad which secured promotion to the Premier League for the first time in 23 years via the playoffs.
Forest are believed to be one of several clubs keeping tabs on the midfielder this summer, while the likes of Tottenham Hotspur, Everton and Leeds United have also been credited with an interest.
However, the Manchester Evening News claims that Southampton have emerged as the frontrunners to sign Garner, who has two years remaining on his contract at Old Trafford.
Garner joined United's academy at the age of eight but has gone on to make only seven senior appearances for the club, including his debut as a 17-year-old in February 2019.
The midfielder, who has since spent time on loan with both Watford and Forest, believes his time in the Championship has helped him to grow as a player and he is now hoping to step up and ply his trade in the Premier League this term.
Speaking to reporters last month, Garner said: "I feel like last season my career has only just started, I'm still young, just turned 21 a couple of months ago. I've had such a good season, with the national team and at club level (on loan at Forest) as well. It's very important now to hopefully play in the Premier League [this] season.
"I think I've come on leaps and bounds. Played a lot of games, 50-plus now, so I've developed a lot physically, which was the question I was asked - whether I'd be able to play, whether I'm ready physically. I think I've shown I'm physically ready to play and, like I said, felt like I've come on leaps and bounds.
"I just feel like, before I went on loan, there were questions of am I physically ready to compete, to play first-team football, and I believe that I've shown that this season. Not only playing a lot of games but how I've held my own in such a tough league and men's football. I think I've shown that."
Garner's potential switch to Southampton would see him become the latest youngster to move to St Mary's, with manager Ralph Hasenhuttl sticking to his model of recruiting highly-rated prospects.
So far this summer, the Saints have signed Gavin Bazunu and Romeo Lavia on permanent deals from Manchester City, while Joe Aribo, Sekou Mara, Armel Bella-Kotchap, and Mateusz Lis have also moved to the South coast.
Should Southampton make swift progress in their pursuit of Garner, the midfielder could be in a position to make his home debut against Man United when the two sides lock horns in the Premier League on August 27.