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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.