Tuanzebe is highly rated at Old Trafford but found opportunities hard to come by last season, starting just nine games.
The centre-back, who has been capped by England at Under-21 level, is keen to continue his development with a full season out on loan and sources have told ESPN that clubs in the Premier League and abroad have already registered interest in signing the 23-year-old for the 2021-22 campaign.
He has a contract at Old Trafford until 2022 with the option of a further year.
Tuanzebe starred in United's 2-1 win at Paris Saint-Germain in the Champions League in October in the absence of captain Harry Maguire but was forced to wait until January for his first league start of the season.
Solskjaer and Tuanzebe are set for talks about his future when the squad returns for preseason training.
Phil Jones is also back in full training after 18 months on the sidelines with a knee injury. The 29-year-old, who still has two years left on his contract plus a 12-month option, has not featured since January 2020.
Tuanzebe has made 37 first team appearances for United since making his senior debut under Jose Mourinho in 2017.
He enjoyed a successful loan spell at Aston Villa between January 2018 and May 2019, helping them win promotion to the Premier League via the playoffs in 2019.