“We are deeply concerned by the disturbing allegations and will continue to closely monitor the situation,” said Nike in a statement.
On Sunday, the Premier League club said Greenwood “will not return to training or play matches until further notice,” after the allegations first emerged.
“Manchester United does not condone violence of any kind,” the statement added.
In a statement sent to CNN on Sunday, Greater Manchester Police (GMP) said “a man in his 20s has been arrested on suspicion of rape and assault,” adding that the suspect “remained in custody for questioning” and that “enquiries were ongoing.”
The police force said it was made aware earlier in the day of “online social media images and videos posted by a woman reporting incidents of physical violence.”
On Monday, GMP said it had been granted “additional time” to speak to a man in his 20s amid the domestic violence accusations.
“Detectives have been granted additional time to speak to a man in his 20s who is being held on suspicion of rape and assault of a woman,” read a statement sent to CNN on Monday.
“The suspect was arrested yesterday (Sunday 30 January) afternoon and remains in custody. Enquiries are ongoing and the victim is being offered specialist support.
“We remind people to avoid any commentary or sharing of images that could compromise the victim’s right to lifelong anonymity, or risk prejudicing a live investigation with active proceedings,” the statement added.
The statement did not confirm either the name of the victim or the potential offender.
This comes after a series of photos and audio recordings allegedly posted online by Greenwood’s girlfriend, social media influencer Harriet Robson.
In the posts, Robson accuses Greenwood of domestic violence, showing photos of what appears to be her lip bleeding and bruises to her body. Robson had taken down the posts on Sunday.
CNN has not independently verified the authenticity of the images or recordings.
Martin Goillandeau contributed to this report.