Schwarzer, now 40 years old, and Karagounis, 36, both have contracts that expire in the summer but Martin Jol has admitted that the club are looking to extend the pair's stay in London.
Despite Damien Duff signing an extension last week, there are still various names in the Fulham squad that could be leaving for free at the end of the season if Fulham don't act. And Jol has made it clear that work is being done to sort it out.
"We are working on it now, constantly, every day because we've got quite a few players who are out of contract," Jol said.
But the Dutchman was prepared to be more specific on Karagounis' situation.
"Giorgio loves to play here and if we feel the moment is right we will sit down with him and offer him a new contract."
It still doesn't answer the questions surrounding Brede Hangeland's future, which Jol did not address. At 31, the defender isn't getting any younger but there's a club-wide acceptance that Fulham would be stronger if he signs a new deal.
There's still little hope that he will do so, though, and with his form improving dramatically over the last few games, the Norwegian could prove to be, by some distance, this summer's most concerning departure. Especially if he goes to Queens Park Rangers.