Russian Foreign Minister Sergei Lavrov and Secretary of State John F. Kerry speak during the International Syria Support Group meeting on Thursday in New York. (Bryan R. Smith/AFP/Getty Images)

Just as Secretary of State John F. Kerry was telling Russian Foreign Minister Sergei Lavrov late Thursday that the only way to salvage Syria’s collapsed cease-fire was to immediately ground the Syrian air force, his chief of staff handed Kerry his BlackBerry.

The Syrian government, Kerry read in a headline, had just announced a new offensive against surrounded rebels and civilians in eastern Aleppo.

An hour later, a weary Kerry told reporters that “the United States will continue to pursue ­every avenue of progress that we can, because it’s the only way to ease the suffering” in Syria. He was “no less determined today than I was yesterday,” Kerry said, “but I am even more frustrated.”

At the closed-door session with Lavrov and representatives of more than a dozen other nations involved in Syria, Kerry said, he had told the group that “the only way” forward was “if the ones who have the air power in this part of the conflict simply stop using it. . . . Absent a major gesture like this, we don’t believe there is a point in making more promises, issuing more plans.”

Kerry’s proposal to ground ­Syrian and Russian aircraft in ­areas of Syria, first made at the U.N. Security Council on Wednesday, was the major topic of conversation in the hour-long session Thursday, one of a series of urgent meetings this week on the margins of the U.N. General Assembly. Russia, said a senior administration official, offered little response.

While making clear after the meeting that he had no substantive progress to announce, Kerry said that “if Russia comes back to us with constructive proposals, we will listen.”

Asked what the United States will do if the Russians do not agree to take “extraordinary steps,” another senior official said that was “something we’re giving a lot of thought to. . . . I don’t think tonight and right now, as we’re approaching a climactic stage of this, is the time to say where we go from here. . . . But that is very much on our minds.” Officials spoke on the condition of anonymity to discuss internal deliberations.

Throughout the day in Aleppo, warplanes suspected of belonging to both Syria and Russia pounded rebel-held areas of the city, with residents and activists describing strikes as intense as they have ever been. Scores of people have been killed in airstrikes since Monday; 21 died Thursday evening when bombs landed on two residential neighborhoods, according to activists living in Aleppo.

“It is a moment of truth for the regime and for the opposition,” Kerry told reporters, “and for all of us who are determined to try to end this war in Syria and defeat the terrorist groups.”

“Kerry’s got the reputation of Mr. Optimist,” an aide acknowledged. But inside the meeting, other members of the group who could “feel and hear the frustration in his voice” urged him not to abandon the effort, the aide said.

Kerry said he would await a response from Lavrov, even as Washington and Moscow continued to exchange charges over responsibility for an attack on an aid convoy in Syria this week that spelled the conclusive end to a partial cease-fire that began just a week earlier. In the one gleam of hope, the United Nations announced that it had made a humanitarian delivery to one besieged town and was prepared to begin several more.

Although they have made a handful of strikes in the past two years against terrorist forces­ that are fighting alongside the rebels, U.S. aircraft have focused on Islamic State targets to the north and east of where the Russians and Syrians have been striking.

“I would not agree that coalition aircraft ought to be grounded,” Gen. Joseph F. Dunford Jr., chairman of the Joint Chiefs of Staff, said in congressional testimony Thursday. “I do agree that Syrian regime aircraft and Russian aircraft should be grounded.”

“There’s no reason to ground our aircraft,” Dunford told the Senate Armed Services Committee. “We’re not barrel-bombing ­civilians; we’re not causing collateral damage.”

Dunford was also asked whether the military was prepared to share “intelligence” with Russian forces­ under the now-suspended cease-fire agreement, provided it could be reinstated and sustained. Assuming a cease-fire and aid delivery proceed, the agreement calls for the United States and Russia to coordinate their counter­terrorism airstrikes.

According to a leaked copy of the terms of the agreement, published Thursday by the Associated Press, it calls for “discussion and sharing of information necessary for the delineation of territories” controlled by the opposition and the Front for the Conquest of Syria, formerly al-Qaeda’s affiliate Jabhat al-Nusra, and sharing “actionable Nusra and Daesh [Islamic State] targets . . . in a way that allows strikes to commence” as soon as the agreement is in place.

“We don’t have any intention of having an intelligence-sharing operation with the Russians,” Dunford said, although he was not asked to distinguish between “intelligence” and the territorial and target “information” sharing called for in the deal.

In its latest statement, the Russian Defense Ministry repeated both its denial of any involvement in the convoy strike, in which at least 20 aid workers and civilians were killed, and its charge that it has proof that an armed U.S. drone was in the area at the time. The Pentagon has denied that it had any aircraft in the area.

A spokesman for the Russian ministry, Maj. Gen. Igor Konashenkov, said that Dunford and Defense Secretary Ashton B. Carter, who also testified Thursday, were expressing “their personal opinions and the fear of responsibility for yet another blunder or deliberate provocation.”

Russia has repeatedly seized opportunities to drive a public wedge between the State Department, which negotiated the deal, and the Pentagon, which has questioned it. In a social-media post Thursday, Russian Foreign Ministry spokeswoman Maria Zakharova seized on Dunford’s statement that he could not say “conclusively” that Russia was responsible for attacking the aid convoy from the air.

Just hours after Kerry had accused the Russians at a Security Council meeting, she wrote, “The Pentagon is shooting him in the back. . . . This is how stupidly and mercilessly they framed the State Department.”

But while Dunford testified that he didn’t know for sure, he said that “what we know are two Russian aircraft . . . were in that area at the time. My judgment would be that they did [strike the convoy]. There were also some other aircraft in the area that belonged to the regime at or about the same time. So I can’t conclusively say that it was the Russians. But it was either Russians or the regime.”

Asked about an internal disagreement, White House press secretary Josh Earnest told reporters in Washington: “There is unanimity of opinion across the U.S. government that there won’t be any cooperation militarily with the Russians unless and until the Russians live up to the commitments they’ve made. There is no disagreement about that.”

In an early Thursday news conference in Geneva, U.N. humanitarian adviser Jan Egeland said that in recent hours the Syrian government had granted permits for aid deliveries that had previously been withheld. Later in the day, a convoy of food and medical supplies arrived in the Damascus suburb of Moadamiya. But no convoys reached Aleppo.

Eastern Aleppo, occupied by the rebels, with 250,000 civilians, “is militarily encircled, it is impossible to get in with our aid,” he said. “Forty trucks are sitting at the Turkish-Syrian border . . . and the drivers are sleeping at the border and had done so now for a week.”

Liz Sly in Beirut, Carol Morello in New York and Missy Ryan in Washington contributed to this report.