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Under intense lobbying from Secretary of State Condoleezza Rice, Israeli and Palestinian negotiators edged toward an agreement early Tuesday on opening a border crossing between the Gaza Strip and Egypt.

Talks extended well into the predawn hours as the two sides debated the central sticking point: how much control Israel would retain over the crossing now that it has withdrawn from the Gaza Strip and handed the territory to the Palestinians. In a sign of the Bush administration’s desire for a breakthrough, Rice delayed her planned departure for Asia to continue talks.

The crossing, at Rafah along Gaza’s southern border, is viewed by Palestinian leaders as an essential gateway for Gaza’s 1.3 million citizens and a potentially crucial port for trade. It is the only direct opening from Gaza to the rest of the Arab world.

Rice met separately Monday morning with Israeli Prime Minister Ariel Sharon and Palestinian Authority President Mahmoud Abbas. After meeting with Abbas, she said the two sides had made headway but not enough. “With some will, and some creativity, agreement should be within sight,” she said.

Palestinian negotiator Saeb Erekat said late Monday that the sides are “very close” and that he was hopeful a solution would be reached.

A senior State Department official who declined to be identified said that, as the negotiations progressed, the issue broadened beyond Rafah to include crossings linking Gaza with Israel that also have been closed or restricted since Israel’s pullout two months ago.

Monday evening, Rice left Jerusalem briefly, flying to Amman to offer condolences to Jordan’s King Abdullah II in the wake of last week’s hotel bombings. After meeting the king and laying a wreath at the Radisson hotel, Rice returned to Israel to continue talks.

Rice said international envoy James Wolfensohn, who has mediated talks on border crossings and other economic issues since the Israeli pullout, had put forward new U.S. proposals to bridge differences between the Israelis and the Palestinians. The proposals called for the presence of European Union monitors at Rafah.

But differences remain over how much control the monitors would exercise and whether Israel should have even indirect oversight of the border through a database of crossers or surveillance cameras.

“We need to augment the Palestinian security weakness without insulting the Palestinians,” said an Israeli government official who declined to be identified because of the subject’s sensitivity. “We know they will stop somebody carrying weapons, but if someone big in a militant group walks across with $50,000 in a satchel, that’s five suicide bombers.”

Tension between the two sides was underscored Monday when Israeli troops killed Amjad Hanawi, the senior commander of the militant group Hamas, in the northern West Bank, even as Rice was in Jerusalem. The Bush administration has opposed Israel’s targeted assassinations.

Wolfensohn has warned about the dire impact of the border closing on the Palestinian economy, with tons of produce rotting in warehouses. In a speech Monday, his frustration seemed to overflow.

“If you want to blow each other up, I have a nice house in Wyoming, and in New York and in Australia and I will watch with sadness as you do it,” he said.