Stekelenburg moved to Roma from Ajax in the summer of 2011 but has struggled to command a first-team place in recent months.
He has fallen behind Mauro Goicoechea in the pecking order at the Stadio Olimpico, prompting him to consider a move elsewhere.
Fulham boss Martin Jol worked with Stekelenburg during his time as Ajax coach in 2009 and has reportedly been monitoring his situation in Italy.
The 30-year-old could be seen as a long-term replacement for current Cottagers No 1 Mark Schwarzer, who turns 41 later this year.
Stekelenburg's agent, Rob Jansen, confirmed that the Dutch star could be making a move to Craven Cottage ahead of the transfer deadline.
"We work very hard to get a deal," Jansen told De Telegraaf.