The 40-year-old's contract expires in the summer and his manager is eager for him to prolong his stay, despite admitting he is looking to bolster his defensive ranks
The 40-year-old 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 after a move last summer, but Jol was coy on other possible targets.
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"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 currently one of three goalkeepers on the books at Fulham alongside Neil Etheridge and David Stockdale, however summer replacements 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 number two so that could be a little problem, so of course we are always looking for a goalkeeper."