Disgraceful abuse containing racist language was directed at the United captain online and it is believed that the posts in question have been reported to the police to investigate.
A statement from the club read: "Manchester United utterly condemns racist social media comments posted around yesterday's Champions League fixture. We are working to identify individuals involved and we will take the strongest possible action we can against them.
"There is no place for racism within our game, or in society as a whole, and we are committed to working to make football free from all forms of discrimination."
A report from the Daily Mail notes that United have been trying to establish whether those involved are genuine 'fans'. It is said that the club is confident these are not people who are season ticket holders or listed in their official supporter database.
Anti-racism group Kick It Out urged Twitter to take stronger action to tackle the problem.
Anti racism campaigners Kick It Out have called on Twitter to do more to stop abuse - after racist posts were put on the site targeting Ashley Young, following United's loss to Barcelona in the Champions League last night #CapitalReports pic.twitter.com/uuIGYkm1jR— Capital Manc News (@CapitalMANNews) April 17, 2019