Fulham boss Jol wants new deal for Karagounis
The Greece midfielder, 36 next month, signed as a free agent earlier this campaign but has impressed Jol.
Jol said: "Giorgios loves to play here. And if we feel the moment is right we will sit down with him and offer him a new contract.
"He's not the youngest. But every time we get other players in, you miss him.
"Last week I didn't play him because I was annoyed because he was playing for the Greece national team, he played 70 minutes and that is not what you want, so I played Emmanuel Frimpong."