QPR loanee Tom Carroll wants to "make it at Tottenham" but is prepared to consider all of his options at the end of the season.
Ilkay Gundogan has commited his future to Borussia Dortmund by signing a new contract.
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.