By the middle of the night in Jerusalem, Israel made its first gestures to Mr. Biden. In a statement, a spokeswoman for the U.S. National Security Council said Israel had agreed to use the Ashdod port to direct aid into Gaza, to open the Erez crossing into northern Gaza for the first time since the Hamas terrorist attack on Oct. 7, and to significantly increase deliveries from Jordan.

Biden administration officials, who spoke on the condition of anonymity to describe the private call in more detail, said that Mr. Netanyahu agreed to additional commitments intended to assuage the president. Among others, the officials said, Israel would promise to institute more measures to reduce civilian casualties and to empower negotiators brokering a temporary cease-fire deal in exchange for the release of hostages.

The reported agreement came as American officials held out the prospect of consequences if Mr. Netanyahu resisted. Secretary of State Antony J. Blinken, who dialed into the call between the president and prime minister, said afterward that Israel needed to do more to increase the flow of humanitarian supplies to Gaza, a challenge that has expanded exponentially since the attack on aid workers prompted some groups to reconsider their activities on the ground.

“If we lose that reverence for human life, we risk becoming indistinguishable from those we confront,” Mr. Blinken said during a stop at NATO headquarters in Brussels. “Here’s the current reality in Gaza despite important steps that Israel has taken to allow assistance into Gaza: The results on the ground are woefully insufficient and unacceptable.”

The secretary of state made clear that the Biden administration was now ready to exact a price if Israel continued to rebuff its counsel. “If we don’t see the changes that we need to see, there’ll be changes in policy,” he said.

The president has long refused to curb the arms flow to influence Israel’s approach to the war. Mr. Biden said after Hamas killed 1,200 people and took hundreds of hostages in October that his support for Israel was “rock solid and unwavering.” While he has increasingly criticized what he sees as the excesses of the military operation, he has until now stuck by his vow.

But with rising agitation on the political left, particularly in electoral swing states like Michigan, even some of Mr. Biden’s closest Democratic allies are coming around to the view that Washington should exercise more control over the weaponry, including Senator Chris Coons, a fellow Democrat from Delaware and confidant of the president.

“I think we’re at that point,” Mr. Coons said on CNN on Thursday morning. If Mr. Netanyahu were to order the Israeli military into the southern Gaza city of Rafah in force and “drop thousand-pound bombs and send in a battalion to go after Hamas and make no provision for civilians or for humanitarian aid,” he added, then “I would vote to condition aid to Israel.”

Mr. Netanyahu did not immediately release a description of his call with Mr. Biden, but in other comments on Thursday he appeared unbowed. In a meeting in Jerusalem with visiting Republican lawmakers organized by the American Israel Public Affairs Committee, known as AIPAC, the prime minister pushed back strongly against Mr. Biden’s longstanding insistence on a two-state solution to the Palestinian conflict.

“There is a contrary move, an attempt to force, ram down our throats a Palestinian state, which will be another terror haven, another launching ground for an attempt, as was the Hamas state in Gaza,” Mr. Netanyahu said. “That is opposed by Israelis, overwhelmingly.”

In a separate video statement, he focused on the threat he sees from Iran. “For years, Iran has been acting against us, both directly and through its proxies, and therefore Israel is acting against Iran and its proxies, in both defensive and offensive operations,” Mr. Netanyahu said, referring to an Israeli airstrike that killed seven Iranian military officers in Syria this week.

“We will know how to defend ourselves,” he added, “and we will operate according to the simple principle by which those who attack us or plan to attack us — we will attack them.”

The White House statement noted that Mr. Biden stood by Israel against Iran during his Thursday call with Mr. Netanyahu, which in addition to Mr. Blinken included Vice President Kamala Harris and Jake Sullivan, the national security adviser.

“The two leaders also discussed public Iranian threats against Israel and the Israeli people,” the statement said. “President Biden made clear that the United States strongly supports Israel in the face of those threats.”

Unlike previous comments, however, the latest White House statement made no mention of Oct. 7 nor the by-now ritual defense of Israel’s right to respond to Hamas. Instead, it emphasized that “an immediate cease-fire is essential” and said that Mr. Biden “urged the prime minister to empower his negotiators to conclude a deal without delay to bring the hostages home.” A person briefed on the situation, who spoke on the condition of anonymity, said negotiators including William J. Burns, the C.I.A. director, will travel to Cairo on Saturday for further talks on such a deal.