The 21-year-old has not figured for the Red Devils since January 2022.
He was suspended by United following allegations of attempted rape, assault and controlling and coercive behaviour.
Those charges were dropped in February, but Greenwood has been working away from the rest of the first-team squad at Old Trafford as an internal investigation is carried out.
A final decision there is said to be close, with it possible that a big call on his future will be made by the end of July – ahead of the 2023-24 Premier League campaign.
Greenwood has not been absorbed back into United’s plans as yet, with senior stars returning from their summer breaks.
The one-cap England international is said to have attracted interest from Turkey and Italy, with a loan move speculated on, and former Red Devils boss Jose Mourinho – who is now in charge of Serie A side Roma – is among those said to be keen.
Greenwood does have a contract in Manchester through to 2025 and was included on United’s retained list at the end of the 2022-23 season.
According to The Athletic, Ten Hag is “open to having Greenwood in his squad at some stage”, but opinion remains divided inside the Red Devils camp.
Chief executive Richard Arnold, along with co-owner Joel Glazer and the rest of United’s board, is expected to make the final ruling on Greenwood’s case and determine whether there is any future for him on the club’s books.