The 40-year-old's contract expires at the end of the season and his manager is eager for him to prolong his stay, despite admitting he is looking to bolster his goalkeeping ranks
The 40-year-old Australia interantional has been in fine form for Fulham, keeping seven clean sheets as the club look to secure a comfortable top-half finish. But his current deal is set to expire at the end of the season.
Fulham have been linked with a move for FC Twente's Nikolay Mihaylov, but Jol was coy on other possible targets.
"I don't deny that we are looking for a goalkeeper," Jol told reporters. "We are not talking with Mihaylov. Last year we made an approach but not this year.
"We are talking to Mark Schwarzer like we did with Brede [Hangeland - who recently penned new terms] and the other players. He is a fantastic goalkeeper so you would have to be very good to fill that space. We have made him an offer so we are waiting. If he accepts he will stay, if not we will talk again like we did with Brede.”
Schwarzer is one of three goalkeepers on the books at Fulham alongside Neil Etheridge and David Stockdale, however off-season signings are inevitable following news that the duo are not content with their second- and third-choice roles at the club.
He continued: "There is a lot of speculation in the press. I have a young goalkeeper in Neil Etheridge, he is out on loan and so is David Stockdale.
"He doesn't want to be No.2 so that could be a little problem, so of course we are always looking for a goalkeeper."