Greater Manchester Police said in a statement that a man was arrested after social media images and videos were posted by a woman reporting incidents of physical violence.
“An investigation was launched and following enquiries we can confirm a man in his 20s has since been arrested on suspicion of rape and assault,” the police said.
The man, who was not named but was described as in his 20s, was being held in custody for questioning, the police added.
The allegations against Greenwood, including video, photographs and a voice note, were posted on Instagram on Sunday morning and later deleted.
“Mason Greenwood will not return to training or play matches until further notice,” United said in a statement to Reuters.
The club earlier said that they “do not condone violence of any kind”.
Greenwood has six goals and two assists in 24 matches in all competitions for United this season. He began his career at United’s academy and has played once for the England national team.