The Manchester United attacker was sent home from England duty on Monday after reports in Iceland claimed he and Manchester City midfielder Phil Foden had met up with local girls in the team hotel and consequently broken strict Covid-19 rules.
The pair were both fined £1,360 for the breach of pandemic rules and both were criticised by England manager Gareth Southgate and their respective clubs.
United wrote in a statement: "Manchester United are liaising with the Football Association and are disappointed with the actions of Mason Greenwood over this situation."
Greenwood flew back to Manchester on Monday after being dropped from the England squad for their game against Denmark and it is understood a decision will be made in due course as to when he can resume training with his club.
Iceland is on the United Kingdom’s travel corridor list which means there is no government legislation which would require the teenager to isolate.
Foden took to his social media channels on Monday to apologise for his actions but Greenwood deleted his Twitter account after the incident and instead issued an apology via the club website on Tuesday morning.
He said: "Having now had the chance to reflect on what’s happened, I can only apologise to everyone for the embarrassment I have caused.
"It was irresponsible of me to break the Covid-19 protocols which are in place to protect players, staff and the public. In particular, I want to apologise to Gareth Southgate, for letting him down, when he had shown great trust in me.
"Playing for England was one of the proudest moments in my life and I only have myself to blame for this huge mistake. I promise my family, the fans, Manchester United and England that this is a lesson I will learn from."
It was Greenwood’s first call-up to the men’s senior side having represented England previously at youth level and he came off the bench in the second half of their 1-0 win against Iceland on Saturday to make his senior debut.
Southgate confirmed the two players had not been in contact with anyone else from the squad after meeting with the girls which meant the game against Denmark was not put in jeopardy.
"These are two boys I don’t know particularly well at the moment so I can’t speak in great depth," said the England boss. "I have got to try and speak with them later in the appropriate way.
"They have got responsibilities and they have apologised. What is clear is that there was a breach of the coronavirus guidelines. We spent such a long time getting those measures in place - the whole squad followed that together.
"It is a very serious situation and we have treated it that way. Obviously, they have been naive, we have dealt with it appropriately."
The FA has launched a "full investigation" into the incident and also apologised to the Football Association of Iceland.