Diplomats and aid workers on Thursday were hammering out the logistics of getting food and medicine into Gaza a day after President Biden said Israel agreed to allow it.

As Gaza grapples with an escalating humanitarian crisis, the prospect of getting aid through the closed Rafah border crossing with Egypt has taken on particular urgency. Trucks have lined up on the Egyptian side, and people hoping to get out have gathered on the Gaza side.

President Biden said Wednesday that he had secured a commitment from Israel’s government to allow food, water and medicine to pass through the crossing and that Egypt had said it would allow 20 aid trucks into the enclave, but did not specify when.

The area around the crossing has repeatedly been bombed by Israel. On Thursday, Egyptian workers were repairing roads so that the large trucks loaded with aid would be able to pass, according to an aid official briefed on the situation.

Hopes are high that the aid trucks would be able to cross into Gaza on Friday, according to European Union officials coordinating aid from the bloc. There are 54 tons of E.U. aid sitting not far from the border in Arish, a city in a restricted area of northeastern Egypt.

Major humanitarian groups including the World Health Organization and the International Committee of the Red Cross also have sent equipment and supplies.

“Our trucks are loaded and ready to go,” Dr. Tedros Adhanom Ghebreyesus, the director general of the World Health Organization, said in a virtual news event. “We’re working with Egypt and Palestine Red Crescent societies to deliver our supplies into Gaza as soon as the border crossing is opened, hopefully tomorrow.”

Egyptian officials are holding intensive talks in Cairo with the U.N. secretary general, António Guterres; his humanitarian chief, Martin Griffiths; and a special U.S. envoy about how to deliver the aid to Gaza.

The border crossing area is closed to non-Egyptian officials. It sits in a province where Egypt has fought militants for years, and the government in Cairo is wary of any spillover from the Gaza conflict.

Credit…Samar Abu Elouf for The New York Times

Previous attempts to secure an agreement that would get aid into Gaza from Egypt have failed to yield concrete results. One sticking point has been over how to ensure that Hamas is not involved in the deliveries or able to divert aid from civilians for its own use.

Officials said Thursday that efforts were underway to address those concerns — most likely by having the Egyptian Red Crescent handle the aid on the Egyptian side and turn it over to the Palestinian Red Crescent on the Gaza side.

Another sticking point is Israel’s demand that the cargo be checked for weapons that Hamas could use for attacks. The American, U.N. and Egyptian officials are discussing who would carry out those cargo inspections, a person directly familiar with the matter said, requesting anonymity to speak about the delicate negotiations.

Hamas, which controls the Gaza side of the crossing, said it had not yet received word to open it up for aid to get through.

“All of Gaza is waiting for the aid,” Wael Abu Omar, the spokesman for the Rafah crossing at Hamas’s interior ministry, said Thursday. “Of course we’ll let the aid through.”

Earlier discussions around opening the crossing had included efforts to get foreign nationals out of Gaza. E.U. officials said on Thursday that there had been no progress on that front. There are approximately 350 to 400 Americans; 1,200 E.U. citizens; 300 Canadians; 200 British citizens and 45 Australians in Gaza, according to diplomats.

As the humanitarian crisis in Gaza intensifies and Israel readies an expected ground invasion, scores of people have been waiting on the Gaza side for days in hopes of leaving.

Abood Okal, a Palestinian American who has been stranded with his wife and one-year-old son, said in a WhatsApp message that he was “very concerned” that there had been no updates about an evacuation despite the talk of an agreement to allow aid into Gaza.

“We feel abandoned to fight on our own for our safety and the safety of our son,” he said in a text message on Thursday.

Mr. Okal said that there had been an explosion not far from the house where he was staying on Thursday morning. With no safe place to go, and no evacuation plan in sight, he wrote, “We are extremely afraid for our lives.”

Iyad Abuheweila contributed reporting from Cairo, and Hiba Yazbek from Jerusalem.