Greenwood will not play for or train at United "until further notice" after being bailed pending further investigation by Greater Manchester Police.
The 20-year-old was arrested on Sunday on suspicion of rape and assault, then further arrested on suspicion of sexual assault and making threats to kill while in custody.
United reaffirmed that Greenwood would not take part in any matches or training following his release on bail, also reiterating their "strong condemnation of violence of any kind".
Having removed merchandise bearing Greenwood's name from their official online store earlier this week, the club have now offered fans the opportunity to swap Greenwood shirts free of charge.
An email sent out to supporters reads: "As a United Direct customer, we are writing to you regarding your previous purchase of a Manchester United shirt personalised with Greenwood, given the current circumstances relating to Mason Greenwood.
"We can confirm that United Direct and adidas customers who purchased a Manchester United shirt that was personalised with Greenwood's name and number, have the opportunity to exchange it for this season's home shirt personalised with another player's name without charge."
Several of United's official sponsors have expressed concern over the allegations, with main shirt partner TeamViewer "closely monitoring developments".
Nike confirmed earlier this week that they have suspended their sponsorship deal with Greenwood, while EA Sports removed the 20-year-old from versions of FIFA 22.
Ralf Rangnick, United's interim manager, insisted that Greenwood's arrest had not been a distraction for the rest of United's players this week but admitted that the "unusual situation" has inevitably been discussed within the group.
"To be honest I didn’t speak that much with the current players about what happened, I’m pretty sure that they did. Obviously we had internal discussions but I’m pretty sure everyone will understand those internal discussions should stay internal," he said.
"It’s an unusual situation for everybody and nobody would want to experience that. It’s an ongoing process and we have to put our focus on tomorrow’s game and then Tuesday’s game at Burnley. This is all I can say.
"Obviously everybody would have wanted those things not to happen but they did and this is what it’s all about in life not only in professional football that you have to stay together and be united on that and still keep going."