Greenwood last played for Manchester United in January 2022, when a was suspended following allegations of rape, assault, and abuse.
The 21-year-old forward was cleared of all charges in February, but has had to work on his own while an internal investigation was ongoing.
A final decision on his future is expected to be made before the end of July ahead of the 2023-2024 season.
Greenwood has attracted interest from Turkey and Italy, where Jose Mourinho is said to be keen on getting him at Roma.
His contract with Manchester United runs until 2025, and according to the Athletic, Ten Hag is ready to have him in the squad at some point, but there are those in the club’s hierarchy who are not as keen.
Manchester United co-owner Joel Glazer, alongside chief executive Joel Arnold and other members of the board are expected to make a final decision on the forward’s future at the club.