Stekelenburg, who has been out of favour for the Italian side of late, held talks with his club on Wednesday as speculation about his future continued to hit the headlines.
It is believed that talks were not hugely constructive, with club officials telling Stekelenburg that his place in the team has all but been secured by current first choice, Mauro Goicoechea.
According to the Metro, Roma are even prepared to sanction a cut price deal for either of the Premier League duo, involving an up front payment of only £3.5 million. Jol has worked with the stopper before, at Ajax, and he is known to be eager to find a long-term replacement for the 40 year-old Mark Schwarzer.
Even so, the Dutch 'keeper's agent, Rob Jansen, has suggested no move is imminent for his client.
"I spoke with [Roma general director] Franco Baldini, and at the moment there is no possibility that Stekelenburg leaves Roma," Jansen told Calciomercato.
"There are no offers from other clubs, but we are at the beginning of January and everything is still possible."