The contracts for both midfielders are due to expire at the end of the season, and they are not being offered fresh deals.
The 36-year-old Karagounis joined the club on a free transfer in September and has played over 20 league games, becoming a firm fans' favourite.
"It is most likely that Sunday will be the last game that Giorgos will play for Fulham," his agent, Paschalis Tountouris, told Goal.
"He's had a terrific year and the Premier league was a great experience. There would be options for him to stay in England and would consider any challenges that may be available.
"It's very good that the fans of Fulham have recognised his contributions this season to vote him as the second best player and he will always have them in his heart."
Davies, 33, is moving on after six successful seasons at Craven Cottage which have seen him play over 150 games - but recently he has been dogged by injury.
"The thing with Simon is as long as I was here he was never fully-fit for more than two or three weeks, and that's a problem," said Fulham boss Martin Jol.
"When he was fit he was a wonderful player but if he's not fit we've got a problem, so we've got to sit down with him and talk about it.
"We will look after Simon and, of course, he can stay here as long as he needs to get fit, but it's about him as well. He has to make his mind up about his own fitness and if he wants to play again."
Fulham have already moved to bring in new midfield reinforcements as they are set to confirm the signing of Ghana international Derek Boateng on a free transfer.