The Red Devils said on Wednesday no decision had been made about the future of the 21-year-old, who has been suspended by the club since January 2022.
The forward was charged with attempted rape, assault, and coercive control nine months later but those charges were dropped in February 2023 due to a combination of the withdrawal of key witnesses and new material that came to light.
After a female United fan took to Twitter to say she would "walk away from my club" if Greenwood returned, Countdown presenter Riley echoed the message.
She told her 684,000 Twitter followers: “I’m with Em, I won’t be able to support United if Greenwood remains at the club. We’ve all seen and heard enough. Pretending this is OK would be a huge part of the problem.
“It would be devastating for my club to contribute to a culture that brushes this under the carpet … I really hope they do the right thing.”
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United were expected to make an announcement on Greenwood's future before their first Premier League game of the season against Wolves on Monday but that game has come and gone.
The club began an internal investigation to decide his future at Old Trafford in February and on Wednesday they said the “fact-finding phase” of the process has finished and the one-time England international's future rests with CEO Richard Arnold.
United added they had gathered "extensive" evidence and context not in the public domain and spoke to numerous people with "direct" involvement or knowledge of the case.
Greenwood, whose contract at Old Trafford expires in 2025, has reportedly been suspended on full pay (approximately £75,000 a week) since his arrest in January 2022.