Egyptian officials say the evacuations were suspended after an Israeli strike on Friday on an ambulance in Gaza that was being used to transport injured people.
Israeli military officials responded by saying the vehicle was carrying Hamas militants, but did not show evidence.
Despite the suspension of evacuations, aid trucks were still able to travel into the territory, two Egyptian sources said.
Another Egyptian official that spoke to Reuters said Egypt's side of the border was open as usual for any evacuees, but that crossings were suspended because of bombardments in Gaza.
Egypt was making "genuine efforts" to resume activity, the official said.
The Rafah crossing to Egypt's Sinai peninsula is the only exit point from Gaza not controlled by Israel.
Speaking on a condition of anonymity, a senior U.S. State Department official said, "we believe it will (open) this afternoon," but adding “don’t hold me to it."
The U.S. diplomat said there had been a "a lot of work" to reopen the crossing over the past 24 hours and thought the problem would be resolved.
"Hamas is making additional demands, and every time that happens, every time one side makes a demand, you have to go work out with every other party involved how to respond," the U.S. diplomat added.
Qatar, which has played a major role in facilitating humanitarian aid and hostage releases, said it was also pushing to resume crossings, but no timeline was set.
"The bombing of hospitals and the bombing of ambulances, which are part of this agreement for ambulance convoys to move patients out of harm's way, certainly does not help," said Majed Al Ansari, spokesperson for Qatar's Ministry of Foreign Affairs in a briefing on Sunday.
"We urge all parties, but especially the occupation army to make sure that there is safe routes and that the agreement, as it was reached, is respected," he added.
The United Nations Secretary General and aid agencies working in Gaza have condemned Israel's air strike on the ambulance.
The Health Ministry, a hospital director and the Palestinian Red Crescent Society in the Hamas-controlled enclave have said the strike targeted a convoy of ambulances evacuating wounded people from the besieged northern Gaza area.
Evacuations began on Wednesday under an internationally brokered deal aimed at letting some foreign passport holders, their dependents and some wounded Gazans out of the enclave.
Unlike on previous days, the Gazan border authority on Sunday did not publish a list of approved foreign passport holders and dependents.
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