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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The start of the 2017-18 Premier League season is just around the corner and is shaping up to be one of the most competitive yet.
As always it is the so called “big clubs” that have splashed the most cash in the transfer window so far this summer. Manchester United or City? Arsenal or Tottenham? Who are the winners and losers of the transfer window so far?