The Premiership club are interested in the Dutch international who is expected to leave the Stadio Olimpico this summer.
Fulham, who initially tried to sign the 30-year-old during the January transfer market, will cost around €3.5m (2.9m).
The Gazzetta dello Sport claims that the deal could be done and dusted within the next 48 hours.
Stekelenburg joined Roma in the summer of 2011 from Ajax for a fee in the region of €6m. He played just 19 League games in 2012-13 after he was dropped by previous boss Zdenek Zeman.
Once Roma sell the custodian, they are then expected to make a move for Rafael Cabral – a 23-year-old ‘keeper who currently plays for Santos.
Roma are also being linked with Emiliano Viviano given the uncertainty over whether Fiorentina will take up their option to sign him outright from Palermo.
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