Secretary of State Antony J. Blinken warned on Wednesday that recent gains in getting desperately needed humanitarian aid to people in the Gaza Strip risked being undone by the fighting in southern Gaza.

For over a week, since Israel began what it describes as a limited military operation against Hamas in the southern city of Rafah, one border crossing vital to the transit of the aid has been closed, and another severely restricted.

“At the very time when Israel was taking important and much needed steps to improve the provision of humanitarian assistance,” Mr. Blinken said, “we’ve seen a negative impact on the fact that we have this active, very active conflict in the in the Rafah area.”

He noted that “we also seen Hamas firing at the crossings themselves, making it also more difficult.”

Mr. Blinken’s comments came as Israel and Egypt traded blame over the continued closure of the crossing between Egypt and Gaza nearly 10 days after the Israeli army took over the Gazan side.

Hundreds of trucks in Egypt have been blocked from reaching Gaza, and wounded and ill people in Gaza who need medical care outside the territory, as well as families trying to escape the war, have been unable to leave.

Humanitarian officials say the closure of the entry point has exacerbated an already devastating aid situation, with prices for many food items surging and fuel becoming scarce.

Israeli forces seized the Gaza side of the Rafah crossing at the Egyptian border as part of what they have described as a limited military operation against Hamas. The raid effectively closed the crossing, though Israeli officials have signaled they are willing to reopen it.

Israel had shut down another crossing — Kerem Shalom — after a Hamas rocket attack nearby killed four Israeli soldiers. It has since reopened, but the aid getting through is still very limited. Egypt, where most of the aid for Gaza is collected and loaded, has resisted sending trucks toward Kerem Shalom, according to multiple officials. American and Israeli officials believe that Egypt is trying to put pressure on Israel to pull back from the Rafah operation.

Israel has accused Egypt of holding up the delivery of aid. Egypt has blamed Israel for the fighting around the border that has created dangerous conditions for aid deliveries to take place. There has been intense bombardment and fighting around Rafah since last week.

Israel Katz, Israel’s foreign minister, said on Tuesday that he had spoken to European foreign ministers about the “need to persuade Egypt” to allow the “continued delivery of international humanitarian aid to Gaza.”

“The key to preventing a humanitarian crisis in Gaza is now in the hands of our Egyptian friends,” he said. He said that Israel would not return the crossing to what he described as Hamas’s control.

Trucks waiting on an Egyptian road along the border with Israel, near the Rafah border crossing with Gaza, earlier this month.Credit…Oren Alon/Reuters

Sameh Shoukry, Egypt’s foreign minister, quickly pushed back, saying that the crossing was still closed because of Israel’s control over it, and because its military operations were putting truck drivers and aid workers in danger. In a statement from the Egyptian Foreign Ministry, Mr. Shoukry rejected what he described as “the policy of twisting facts.”

Israeli officials rarely make public criticisms of Egypt, with which Israel maintains a decades-old peace treaty and sensitive security cooperation.

Defying international pressure, Israel launched a limited invasion of Rafah on May 6, taking over areas in the east. Even if trucks were allowed to pass through the Rafah crossing, it was not clear if they could safely navigate through eastern Rafah, where Israeli forces have been fighting Hamas militants.

Israel recently opened two new routes for aid trucks to enter northern Gaza directly.

“We’re also seeing real progress in the north, where more is coming through,” Mr. Blinken said. “But what we don’t want to see is a situation where we basically reverse what’s happened in recent months.”

Palestinians workers evacuated the Kerem Shalom crossing before the arrival of Israeli forces, according to Wael Abu Omar, a spokesman for the Palestinian side of the crossing. Israel has asked the Palestinian Authority to send its employees to help run the crossing, but not in their official capacity, said two Palestinian officials, who spoke on the condition of anonymity to discuss messages exchanged between Israeli and Palestinian authorities.

The authority’s leadership swiftly rejected the proposal, said the officials.

Mahmoud al-Habbash, a religious affairs adviser to Mahmoud Abbas, the president of the authority, said that Israel needed to withdraw from the crossing before Palestinian Authority employees would take responsibility for operating it.

“How can we work with Israeli tanks there?” he asked in an interview. “In terms of principle, that’s unacceptable, not to mention the fact that would be dangerous for Palestinians.”

The Ramallah-based Palestinian leadership, Mr. al-Habbash said, would also need to be reassured that the return of the authority to the Rafah crossing was part of a broader effort to reintegrate the governing body back into Gaza.

“We don’t reject steps being taken one after another, but it must be known that the Rafah crossing is part of Gaza, and a solution for the Rafah crossing is part of the solution for Gaza, and the solution for Gaza is part of the solution for all parts of the Palestinian state,” he said.

In public statements, Prime Minister Benjamin Netanyahu has all but ruled out the authority returning to Gaza, a prospect supported by the United States.

Aaron Boxerman contributed reporting.