The source told the Qatari Al-Sharq Saturday that the request was conveyed by Netanyahu's national security advisor, Uzi Arad, who visited Cairo on Thursday.
The Hamas official said Israel was interested in spurring talks on the matter for two reasons: "In order to decrease the pressure on the prime minister on the home front from those who demand that he release Shalit, and to divert attention from Israel's efforts to build 50,000 housing units in the West Bank to be used as a barrier between Hebron and the rest of the area."
He said Cairo had not yet appealed to Hamas to renew the indirect talks on the captive soldier, but stressed that negotiations between the organization and Egypt were still ongoing.
The official added that Hamas had not altered his demands since they were declared when the talks began, and that it would not be reasonable to expect that after four years of demanding the release of 450 prisoners the organization would withdraw its demands and agree to a deal he called "more of a deportation deal than a prisoner swap deal".
The Palestinian Al-Quds reported Friday that during Arad's short visit to Cairo he met with a number of Egyptian officials on matters including the prisoner swap with Hamas.
In addition, dozens gathered outside Netanyahu's home in Caesarea Friday evening to demand Shalit's release.
The protestors held an evening prayer service to welcome the incoming Shabbat (Kabbalat Shabbat) and sang songs. Some of them waved flags bearing Shalit's image.
"It cannot be that the prime minister is sitting at home and Gilad Shalit isn't," said Magen Dahari, one of the organizers.
"The holiday of freedom (Pesach) is approaching, and Netanyahu is free while Gilad is not. We call on Netanyahu to secure Shalit's release, because he (PM) is the one who can free him," Dahari told Ynet. "Bibi, it is your responsibility to bring Gilad home."