Hamas had demanded that Egypt reopen the Rafah crossing to allow the pilgrims to pass directly into the coastal territory rather than force them to pass through Israeli-controlled Kerem Shalom border crossing.
Israel and Egypt both insisted however that they all pass through Israeli security checks, fearing Hamas militants hiding amongst them would try to smuggle in large sums of cash into Gaza.
Palestinians sources claimed that the agreement stipulated that Egypt would check the Palestinians and report any suspected smuggling incidents to Israel.
Hamas blamed Israel and the PA government, led by Palestinian President Mahmoud Abbas, for trying to use the pilgrims to leverage political pressure on the organization.
President Hosni Mubarak had said on Sunday that he wanted to see a quietly negotiated solution.
"Those Palestinians are our brothers. We'll find a solution for them, but let's do without loud mouthing. Negotiations won't work that way," he told a news conference in Cairo.
The case of the pilgrims gave rise to a heated debate in the Egyptian parliament on Sunday, with most members favoring their return directly to Gaza without Israeli checks.
Hamas Islamists called on Egypt to open its shuttered border crossing with the Gaza Strip to let the Palestinians return to their Gaza homes on Saturday.
Hamas officials estimated that 2,200 Gaza pilgrims were stranded on ships at an Egyptian port on the Red Sea.
"We are aware of the Israeli and American pressures on Egypt, and we urge Egypt to reject these pressures and to allow the pilgrims a safe return through Rafah," Hamas official Sami Abu Zuhri told a news conference.
Reuters contributed to this report