Jordan Pickford has been recognised as England's number one for the past few years but Pope, 29, started all three games in the recent international break.
Burnley manager Sean Dyche gave an upbeat assessment on the 'keeper, who missed the last three games of the Premier League season.
"We're expecting the recovery to go well at this early stage, but we'll wait and see as he recovers," he said. "He's had a really, really good season once again. He continues to develop into a top, top 'keeper. He's incredibly unfortunate to have this happen at this time but once again he's shown what a top 'keeper he is during the season and I'm sure will do again going into next season once he recovers."
Southgate said: "For Nick, it's heartbreaking to get an injury so close to a major tournament. He's been a fantastic member of our squad, a totally unselfish guy."
Standout matches today