Icelandic media outlet DV reported that the two players were visited by two women in their hotel.
“Unfortunately this morning [it] has been brought to my attention that two boys have violated Covid guidelines regarding our safety bubble, ”Southgate told reporters on Monday.
“We had to decide very quickly that they couldn’t have any interaction with the rest of the team and wouldn’t be able to travel to practice.
“Given the procedures we have to follow now, they will have to return to England separately.”
England players have been urged to stay in a secure squad bubble during the Nations League matches.
Manchester City’s Foden and Manchester United’s Greenwood won their first caps in Saturday’s game, with Foden playing the full 90 minutes while Greenwood stepped in as striker Harry Kane’s replacement in the second half.
Both players were kept away from breakfast and training on Monday after it emerged they had broken protocols, according to Southgate, who added that no other squad member ‘had been in contact with them.
“Nothing has happened in the areas we occupy in the hotel,” he said. “We are still in the process of taking stock of all the information because, as I said, it was only brought to my attention a few hours before training.
“No one outside of our group has been in the areas of the hotel we occupy.”
He later added: “Right now I’m trying to collect a lot of information. Obviously, they were naïve. We treated it appropriately.
“I think of everything from there, I recognize their age, but of course the whole world is facing this pandemic and there is the responsibility of every age group to make sure they play their part to protect each other and not spread the virus.
“We certainly don’t underestimate how important it has been to try and maintain this bubble the way we have.”
England beat Iceland 1-0 after a chaotic end to the game which saw Raheem Sterling score a late penalty before Birkir Bjarnason was sent off from the penalty spot for the host team in the stoppage time.
England will also be without defender Kyle Walker for the trip to Denmark, who lost 2-0 to Belgium in the other group game, as he is serving a suspension for being red carded in Saturday’s game.
Manchester City said Foden’s actions were “wholly inappropriate”.
A club statement continued: ‘His behavior not only directly contravenes strict guidelines related to Covid-19, but also falls well below the level expected of a Manchester City player and an England international.
“The club are supporting the FA regarding this incident, and club officials are now in contact regarding Phil’s early forced return to the UK.”
CNN reached out to Manchester United for comment but did not immediately receive a response. In a statement sent to Manchester Evening News, the club said it “had relations with the Football Association and was disappointed with Mason Greenwood’s actions in this situation”.