The 21-year-old is currently suspended following charges of rape, assault and controlling and coercive behaviour, handed down in January 2022. These charges were dropped in February 2023.
While United have continued their own investigation into matters, a decision is yet to be made on Greenwood's future.
Given the nature of the accusations against him, a return to United seemed highly unlikely. However, a report from The Sun claims that manager Ten Hag and 'most' of Greenwood's team-mates would welcome the decision.
The Dutchman was previously quoted as saying Greenwood had "scored goals in the past" for United.
The Sun adds that any decision on the player's future is set to reside with the club's American owners, the Glazers, and that a decision could be made within the next week. There are fears over the backlash - not least with commercial and sponsorship deals - that Greenwood's reintegration would cause.
Indeed, it was revealed last month that the decision will be strongly influenced by the opinions of the club's stakeholders. United's women's side have previously raised their concerns over the forward's potential return.
Greenwood, according to The Sun, is impatient for the club to make a decision, putting an end to a saga that has lasted over 18 weeks, and is said to be "delighted" that some of the squad and Ten Hag would reportedly welcome him back.
Rumours of a potential loan spell to Spain, Italy or Turkey were mooted on Thursday. United have yet to make any public statement on Greenwood since they "noted" charges being dropped earlier in the year.