The 29-year-old will have surgery next week, his club said, and will be assessed to see if he will make England's squad for the tournament, which is named on Tuesday.
"Unfortunately, Nick is going to need a minor operation on his knee this week," Burnley manager Sean Dyche told BT Sport, as he was forced to leave Pope out of the season-ending clash with Sheffield United.
"That's not ideal for him or us and possibly for the England team. We'll wait and see. It's nothing too serious, we don't think, but it still needed doing.
"He was trying to get right for this game to give it a test but didn't quite come through in training, so he and we have had to make a judgement call on that.
"It's mainly down to him and he knows what he's got to do. He'll have that (the operation) done this week and hopefully it will settle down for a quick recovery."
Despite Jordan Pickford being recognised by most as England's number one, Pope was the man in possession of the gloves having started all three games in the most recent international break.