Klopp is quoted as saying "now we have to make decisions" and "negotiations need to be done" in the post, but as of yet there has been no official word from the club on deals that could be close to completion.
The 25-year-old, who now plays for Swedish side AFC Eskilstuna, said he didn't think the season had gone well for his boyhood club but missing out on a place in Europe's top club competition could be a catalyst for positive change.
He said his future would be decided after next week's FA Cup final against Chelsea and that his loyalty could not be questioned, adding that he had turned down "every club in the world" to stay at Arsenal for 21 years.
"That's the competition that exists now in England and I think Everton will be on the scene".
Liverpool playmaker Philippe Coutinho has reassured Liverpool fans that he has no intention of leaving this summer to become "just another" player at one of Europe's big clubs.
He added: "We do our job and you are professional and part of the job is being professional when the environment is not positive". "The players came back stronger in the past two months and I'm very proud of them for doing that".
Wenger did not give any further hints on his immediate future, insisting he did not know whether the cup final would be his last match in charge of Arsenal.
"I do adhere to the fact that some young players want to play Champions League and stuff, but the money is what gets players through the door, and Liverpool pay and Liverpool will pay". They managed to get points and we had a few results which didn't go our way.
Klopp said: "We've had bigger games, but this was the game of the season though".
Much more is expected this time round and, before flying to Australia, Klopp outlined his transfer strategy for the summer, which will be key to improving their Champions League record of just three wins in 14 games since the 2008-09 quarter-finals.