According to the Manchester Evening News, Tottenham and Nottingham Forest are both interested in 21-year-old Garner, who is valued between £15 million and £20 million by United.
Garner has a contract with United through 2024.
Garner started the friendly match against Rayo Vallecano at Old Trafford but only played 22 minutes overall during the club's preseason tour of Thailand and Australia due to injury.
Due to the club's inability to sign a central midfielder, the decision to sell Garner is likely to spark discontent among United supporters.
Scott McTominay and Fred were both started by United manager Erik ten Hag in the 2-1 loss to Brighton; McTominay's performance was heavily criticized. Fred was replaced after 53 minutes. Ten Hag then benched McTominay for the defeat to Brentford but substituted Fred for him with United trailing 4-0 at the break.
Former United manager Jose Mourinho championed Garner, including including him in the pre-season tour squad of the United States in 2018 and starting him against the San Jose Earthquakes.
Garner made his debut under Ole Gunnar Solskjaer at Crystal Palace in February 2019 and has appeared seven times for the first-team.
The England Under-21 international enjoyed a superb loan spell at Forest last season and helped them return to the Premier League for the first time in 23 years.
United’s failure to reinforce their central midfield was exposed by Brighton and Brentford, and their inordinate pursuit of Frenkie de Jong is nearing the 100-day mark.
The clubs have agreed a fee of £15m rising to £20m for Juventus midfielder Adrien Rabiot and are continuing negotiations with the Frenchman’s mother, Veronique.