Midfielder Klaassen and striker Ramirez have not travelled with the squad to Croatia for Thursday's play-off second leg against Hajduk Split.
Everton manager Ronald Koeman confirmed at his pre-match press conference that Klaassen is sidelined due to a foot injury while Sandro remains troubled by a heel problem.
"Sandro did not travel and neither has Davy Klaassen, who got a knock on his foot," Koeman said.
"Davy will be out for at least one week but he will be fit after the international break. Sandro is a doubt for Sunday's game (against Chelsea)."
Everton hold a 2-0 lead from the first leg after goals from Michael Keane and Idrissa Gana Gueye at Goodison Park.
Koeman will decide on Thursday whether to start the club's record signing Gylfi Sigurdsson.
Sigurdsson, who made his debut as a substitute against Manchester City two days ago, will definitely be involved in Croatia.
"I need to make the decision," Koeman added.
"He got game-time on Monday, and now the decision is whether he starts or comes on during the game. We have one more session then I will make that decision on Thursday morning."
Wayne Rooney is set to start for Everton - his first appearance since retiring from international football - and Koeman said: "He had a talk with the national coach and made his decision because he thinks it is the best for Everton.
"It is always the player's decision and, of course, I respect that.
"If you reach the group stage of the Europa League and have international duty, then that is a really tough programme.
"His fitness is really good at the moment. It is up to the player to decide the best thing to do.
"I respect that and I would have respected it if he had made a different decision about his future as an international player.
"I think it is a good decision for him and a good decision for Everton. It means when the internationals come, he will have some time at home with his family."
Koeman confirmed James McCarthy's absence due to a knee complaint while Ramiro Funes Mori, Yannick Bolasie (both knee), Seamus Coleman (leg) and Ross Barkley (hamstring) remain absent.