"He has been offered a new deal at the club which I hope he will be happy with and accept."
On Schwarzer, Hodgson added: "Mark will be going into the last year of his contract and that is nothing which either he or we want so I'm hoping that will be quickly resolved.
"It's all very well talking about possible reinforcements and strengthening of your team but this group of players here this year have done a great job for us and it's important for me that we keep that group together."
One player Hodgson will not keep is on-loan Giles Barnes, who will return to Derby and faces surgery on his Achilles tendon.
The Fulham boss also confirmed that 34-year-old France international midfielder Olivier Dacourt, who arrived from Inter Milan in February, is leaving Craven Cottage.
"Olivier came here making it perfectly clear he would be here until the end of the season and I think he's contemplating retirement now."
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