The Cottagers have been linked with the Dutch stopper in recent weeks, with manager Martin Jol keen to strengthen his squad.
Current number one goalkeeper Mark Schwarzer is now 40-years-of-age, and back-up David Stockdale has left to join Hull on loan, tempting the former Tottenham coach to search for fresh options.
Stekelenburg is believed to be available, having lost his position in the first-team with Roma.
But, Real Madrid are now said to be interested, with Iker Casillas having been ruled out for over a month through injury.
Reserve stopper Antonio Adan will fill in, but manager Jose Mourinho is thought to be unconvinced by his abilities.
This could force him into the transfer market, with Stekelenburg a viable option.
They have been linked with a move for QPR’s Julio Cesar, but Harry Redknapp is thought to be unwilling to allow the Brazilian to leave this month.
Real’s interest will come as a blow to Fulham, who may turn their attention to Birmingham’s Jack Butland.
The 19-year-old England international is believed to be available for around £6m, but the London club would face competition from Premier League rivals Liverpool and Southampton for his signature.
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