Stockdale, who has been recently sent out on loan to Hull City, was thought to be out of favour at Craven Cottage and rumours have recently been circulating that Martin Jol has opted to place the goalkeeper on the transfer list, alongside Matthew Briggs.
Stockdale quickly moved on Twitter to quell the speculation and has recently revealed, to the Daily Mirror, that the club are looking to him to replace Schwarzer when the Australian eventually hangs up his gloves.
"It's a terrific opportunity for me," said the Yorkshireman.
"I have been struggling to get games in the Premier League because we have been doing well at Fulham.
"They want me to stay and hopefully, take Mark's gloves when he has finished.
"If I do well here, you never know what could happen in the future."