Icelandic media outlet DV reported the two players were visited by two women in their hotel.
“Unfortunately, this morning [it] was brought to my attention that two boys had broken the Covid guidelines in terms of our secure 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 training.
“Given the procedures that we have to follow now they’ll have to travel back to England separately.”
England players have been told to stay in a secure team bubble during Nations League games.
Both Manchester City’s Foden and Manchester United’s Greenwood won their first caps in Saturday’s game, Foden playing the full 90 minutes while Greenwood came on as a second-half substitute for striker Harry Kane.
The two 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 team members had been in contact with them.
“Nothing has happened in the areas that we occupy in the hotel,” he said. “We’re still getting to the depths of all the information because, as I said, this was only brought to my attention a couple of hours before training.
“Nobody from outside our party has been into the areas of the hotel that we’re occupying.”
He later added: “At the moment I’m trying to take a lot of information in. Obviously, they’ve been naïve. We’ve dealt with it appropriately.
“I think everything from there on, I recognize their age but of course the whole world is dealing with this pandemic and there’s responsibility on every age group to make sure that they play their part in keeping each other safe and not spreading the virus.
“We’re certainly not underestimating how important it’s been to try and keep that bubble as we have.”
England defeated Iceland 1-0 after a chaotic end to the game that saw Raheem Sterling score a late penalty before Birkir Bjarnason misfired from the penalty spot for the host side in stoppage time.
England will also be without defender Kyle Walker for the trip to Denmark, which lost 2-0 to Belgium in the group’s other game, as he serves a suspension for being red carded against in Saturday’s game.
Manchester City said Foden’s actions were “totally inappropriate.”
A statement from the club continued: “His behaviour not only directly contravenes strict guidelines related to Covid-19, but also falls well below the standard expected of a Manchester City player and England international.
“The club supports the FA regarding this incident, and officials from the club are now in touch in relation to Phil’s enforced early 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 is “liaising with the Football Association and are disappointed with the actions of Mason Greenwood over this situation.”