The Liverpool right back picked up the injury in the second half of England’s 1-0 friendly win over Austria on Wednesday.
Alexander-Arnold was one of four right backs in the England squad and his replacement will be made known after Sunday’s friendly game against Romania.
"Well it's heartbreaking really, for any player to get to the eve of a major tournament, be named in the squad and then to miss out through injury," Southgate told a news conference.
"You know how rare these opportunities are, even though he's a young player who's going to have these opportunities again, that's a really difficult moment for him especially.
"It's a big disappointment for us of course, as well, but you can't help, first and foremost, to feel for Trent in this situation.
"The one thing I was pleased about immediately after the game, I didn't like the look of the injury but at least it sounds like he'll be fine for pre-season and next season, but that is a very small positive. Of course he was very upset, bitterly disappointed and in those moments everybody is thinking about him."
Alexander-Arnold has won 13 caps for the England national team, scoring one goal.
