The 29-year-old has been linked with a move away from Roma in recent months.
Stekelenburg only joined the Italian giants from Dutch side Ajax last summer.
But Fulham are confident of signing the goalkeeper, who has previously been linked with the likes of Manchester United and Arsenal.
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.