After circling each other for the past year, President Obama and President Putin squared off on Monday at the United Nations in dueling speeches that presented starkly different views on the Syrian crisis and how to bring stability to the Middle East.

President Obama made a forceful defense of diplomacy and the system of rules represented by the international body, but in a veiled reference to Mr. Putin, he warned that “dangerous currents risk pulling us back into a darker, more disordered world.”

Mr. Putin talked about mounting a broad effort to support Syria’s president, Bashar al-Assad, as the best bulwark against the spread of the Islamic State and other radical groups, even though the White House has said Mr. Assad has to leave power if there is to be a political solution in Syria.

Beyond the verbal jousting and steely looks over lunch after the morning speeches, however, the two leaders were still playing a subtle game of diplomatic poker, each trying to maneuver the other into shifting his position. 

For the White House, this has meant accepting a Russian role in the region but hoping that Moscow will appreciate the risk of becoming bogged down. That, they hope, will raise the costs of backing Mr. Assad and force Russia to work sincerely on a political transition that will lead to the Syrian leader’s departure.

“Knock yourselves out,” one Obama administration official said, mocking Mr. Putin’s bravado about forming a grand coalition in Syria.

For the Kremlin, it means restoring enough stability to Syria to win acceptance of an expanded role for Russia in the Middle East — not to speak of its expanded military presence. Such a development, in the Kremlin’s view, would also validate Mr. Putin’s contention that toppling authoritarian governments in the Middle East has led only to chaos and sanctuaries for terrorists.

Two speeches, one reception and a meeting later, there was no hint that the two leaders had substantially narrowed the chasm between them on their principal disagreement: the future of Mr. Assad. 

“The Obama administration would like to find a way to link arms with Russia on a diplomatic process and not have to tackle some of the less palatable issues like creating safe areas in Syria,” said Andrew S. Weiss, vice president for studies at the Carnegie Endowment for International Peace. “But the only road map Putin laid out today was a fuzzy concept of a grand coalition to fight terrorism arm in arm with Bashar al-Assad, the very man the Americans say is the source of the problem.”

After the Russians surprised the Obama administration by deploying warplanes, tanks and marines at an airfield near Latakia, Syria, the White House agreed to hold military-to-military talks to ensure against any accidents leading to a confrontation. But the larger hope, as Secretary of State John Kerry made clear on Sunday, was that the two sides might work out a common political strategy on Syria. 

There was no hint of that in the two leaders’ speeches on Monday.

Mr. Obama singled out Russia’s annexation of Crimea as a flagrant violation of the international order. On Syria, he repeated the administration’s insistence that Mr. Assad would ultimately have to step down, though he provided no clues as to what steps the United States might take to pressure him to hand over power. 

“The United States is prepared to work with any nation, including Russia and Iran, to resolve the conflict,” Mr. Obama said. “But we must recognize that there cannot be, after so much bloodshed, so much carnage, a return to the prewar status quo.”

Mr. Obama also talked about a “managed transition” in Syria, in which Mr. Assad would be gradually eased out of power. There are intense discussions underway on how long that transitional period should be and how many in Mr. Assad’s close circle would have to go, several United Nations Security Council diplomats said.

Mr. Putin, who was making his first appearance at the United Nations General Assembly in 10 years, was openly dismissive of the United States’ interventions in the Middle East. The United States-led effort to oust Saddam Hussein in Iraq and Col. Muammar el-Qaddafi in Libya, he said, had made each country a haven for terrorists.

And the Obama administration’s attempts to train and equip a moderate Syrian opposition would end up swelling the ranks of Islamic radicals, Mr. Putin insisted. The Kremlin says about 2,000 of the extremists who have joined the Islamic State have come from Russia, fueling concern that they may return and carry out terrorist attacks. Russia has fought two wars against Islamist separatists in Chechnya.

Better, Mr. Putin said, to rally around Mr. Assad. “We think it is an enormous mistake to refuse to cooperate with the Syrian government and its armed forces, who are valiantly fighting terrorism face to face,” he said.

But he offered no prescription for how the Syrian political crisis might be resolved. Nor did Mr. Putin indicate that the Russian military buildup in Syria — including its first new military base in the Middle East in decades — would be reversed if the Islamic State was defeated.

Russia and Syria are longtime allies, with deeply interwoven personal and military connections that militate against any wholesale abandonment of Damascus by the Kremlin.

“The Russians have consistently said that they are not attached to Assad personally, but they insist his government is legitimate, that it is fighting terrorists, and they have rejected efforts from any outside country or combination of countries to impose a particular person or formation to replace Assad,” Robert S. Ford, a former American ambassador to Syria, said in a recent interview. “And they have stressed they want to preserve the Syrian state and its institutions, many of which it has long had close ties with.”

On Monday evening, the two presidents entered a small room with Russian and American flags and shook hands before their widely anticipated meeting. They ignored shouted questions on Syria. The meeting, the first between the two leaders in two years, was held in a Security Council consultation room.

Before the closed session, the prospects for close cooperation did not appear auspicious. Mr. Putin did not provide notice to Mr. Obama of the Russian decision earlier this month to set up an air hub near Latakia or to conclude an intelligence-sharing agreement on Sunday with Iraq, Iran and the Syrian government.

In recent days, the two sides sparred even over which one wanted the meeting more. Mr. Putin also appears to be coordinating his strategy with Iran, which has been Mr. Assad’s strongest backer.

Still, Mr. Obama and Mr. Putin did manage to work together in 2013 to forge an accord that called for Syria to give up its chemical arsenal. Mr. Putin pursued that accord to head off an American military strike that might have emboldened the Syrian opposition and undermined Mr. Assad, and since then the conflict in Ukraine has soured relations.

After the meeting Monday night, Mr. Putin said the discussions had been “very constructive, businesslike and frank.” American officials, who insisted on anonymity as a condition of briefing reporters, echoed that description, noting that half of the session had been spent on Ukraine and half on Syria.

Still, there was nothing to suggest that the two sides had overcome their differences on the future of Mr. Assad. “I think the Russians certainly understood the importance of there being a political resolution in Syria and there being a process that pursues a political resolution,” an American official said. “We have a difference about what the outcome of that process would be.”

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