Hodgson has managed to guide the Cottagers to the Europa League final this season.
The 62-year-old has been strongly linked with the job as Liverpool manager in recent days.
Schwarzer said: “Would winning the Europa League persuade him to stay? I don’t know, it might work the other way.
“Whenever there’s success, whenever someone has achieved what our gaffer has in the last few years, he’s always going to be linked with other positions.
“Obviously he’s had a lot of help from everyone else together – but, inevitably, he’s the man in charge and he makes the decisions.
“He’s been extremely successful and we just hope as players and as a club that we can hold on to him as long as possible.
“I think it’s a truly remarkable season and the manager, for me, without a doubt is the Manager of the Season.”
