Greenwood is waiting on the club to make a call regarding his future as an internal investigation is still going on.
The 21-year-old has not featured for United since he was arrested in January 2022 on suspicion of rape and assault. He was suspended by the club and later charged with attempted rape, assault occasioning actual bodily harm, and controlling and coercive behaviour.
The charges against Greenwood were dropped but he has still not been welcomed back into the fold by the Red Devils.
According to The Sun, United are set to make a final decision on Greenwood's place at the club before the end of the month, and Ten Hag is waiting on the verdict after giving his opinion.
"On this, it’s a club decision and of course, I said my opinions but it's a club decision and we have to accept that," the United boss told reporters before his team's latest pre-season friendly against Arsenal.
"I can't say so much about it but I shared my opinion about it, so let's see what's going to happen."
Greenwood has been linked with a move away from United, with Juventus said to be interested in taking him on loan.
Roma have also expressed interest in him and it has been reported that he could be sent to Atalanta as part of the proposed deal to bring Rasmus Hojlund to Old Trafford.