WASHINGTON — The United States accurately predicted the start of the war in Ukraine, sounding the alarm that an invasion was imminent despite Moscow’s denials and Europe’s skepticism. Predicting how it might end is proving far more difficult.

There are three separate back-channel efforts underway to start negotiations — by the leaders of France; Israel and Turkey; and, in a recent entree, the new chancellor of Germany. But so far, all have hit the stone wall of Russian President Vladimir V. Putin’s refusal to engage in any serious negotiation. At the Pentagon, there are models of a slogging conflict that brings more needless death and destruction to a nascent European democracy, and others in which Mr. Putin settles for what some believe was his original objective: seizing a broad swath of the south and east, connecting Russia by land to Crimea, which he annexed in 2014.

And there is a more terrifying endgame, in which NATO nations get sucked more directly into the conflict, by accident or design. That possibility became more vivid on Sunday, when Russian missiles landed in Ukraine’s western reaches, an area unscathed until now by the 18-day-old conflict, about a dozen miles from the Polish border. Russia declared over the weekend that continued efforts to funnel weapons through that region to the Ukrainian forces would make the convoys “legitimate targets,’’ a warning that just because the weapons are being massed on NATO territory does not mean they are immune from attack.

In interviews with senior American and European officials in recent days, there is a consensus on one point: Just as the last two weeks revealed that Russia’s vaunted military faltered in its invasion plan, the next two or three may reveal whether Ukraine can survive as a state, and negotiate an end to the war. So far even the most basic progress, such as establishing safe humanitarian corridors, has proved elusive.

And now, what troubles officials is that Mr. Putin may double down and expand the fight beyond Ukraine.

In private, officials express concern that Mr. Putin might seek to take Moldova, another former Soviet republic that has never joined NATO and is considered particularly vulnerable. There is renewed apprehension about Georgia, which fought a war with Russia in 2008 that today seems like a test run for the far larger conflict playing out.

And there is the possibility that Mr. Putin, angered by the slowness of his offensive in Ukraine, may reach for other weapons: chemical, biological, nuclear and cyber.

Jake Sullivan, Mr. Biden’s national security adviser, mentioned that scenario on Sunday, appearing on CBS’s “Face the Nation.” “Part of the reason why Putin is resorting to the possibility of extreme tactics like the use of chemical weapons is because he’s frustrated because his forces aren’t advancing,’’ he said.

Mr. Sullivan said that Russia would suffer “severe consequences” if it used chemical weapons, without specifying what those would be. He sidestepped the question of how Mr. Biden would react. So far he has said the only thing that would bring the United States and its allies directly into the war would be an attack on NATO nations. Quietly, the White House and the senior American military leadership have been modeling how they would respond to a series of escalations, including major cyberattacks on American financial institutions and the use of a tactical or “battlefield” nuclear weapon by Mr. Putin to signal to the rest of the world that he would brook no interference as he moves to crush Ukraine.

Even with Ukrainians begging for more offensive weapons and American intervention, Mr. Biden has stuck to his determination that he will not directly engage the forces of a nuclear-armed superpower.

“The idea that we’re going to send in offensive equipment,” Mr. Biden said in Philadelphia to the House Democratic Caucus on Friday, “and have planes and tanks and trains going in with American pilots and American crews, just understand — and don’t kid yourself, no matter what you all say — that’s called ‘World War III.’ OK? Let’s get it straight here.”

Diplomacy: Deciphering Putin’s Bottom Line

Early last week there was a glimmer of hope that a real negotiation would begin that could establish humanitarian corridors for Ukrainians to escape the horror of intense shelling and missile attacks, and perhaps lead to peace talks. Dmitri Peskov, the Kremlin spokesman and a confidant of Mr. Putin, said that if Ukraine changed its constitution to accept some form of “neutrality” rather than an aspiration to join NATO; recognized that the separatist areas of Donetsk and Lugansk were independent states, and that Crimea was part of Russia; the military strikes would stop “in a moment.”

In an interview with ABC News the next day, President Volodymyr Zelensky of Ukraine seemed surprisingly open to the idea. He said he had “cooled down” on joining NATO, saying it was clear the Western alliance “is not prepared to accept Ukraine.’’ And while he did not say he could accept a carve-out of part of the country, he said that “we can discuss and find a compromise on how these territories will live on.”

But it is unclear whether Mr. Putin himself would take that deal. Separate conversations between the Russian leader and President Emmanuel Macron of France, Chancellor Olaf Scholz of Germany, Prime Minister Naftali Bennett of Israel and President Recep Tayyip Erdogan of Turkey all circled the same issues, but left his interlocutors wondering if they were being played for time as the war ground on.

A French government account of a call to Mr. Putin on Saturday by Mr. Macron and Mr. Scholz termed it “disappointing with Putin’s insincerity: He is determined to continue the war.” Wendy Sherman, the deputy secretary of state, said there was no evidence from the conversations so far that Mr. Putin has changed course; he remains “intent on destroying Ukraine.”

Each of those leaders checked in with senior U.S. administration officials before and after their talks with Mr. Putin, and they have been speaking with Mr. Zelensky as well. The United States has kept some distance — in part because no senior Russian officials will communicate with their American counterparts, including with the kind of talks that were routine in the run-up to the war.

The best hope, American and European officials say, is that Mr. Putin concludes that he must scale back his goals in the face of the economic sanctions — especially the crippling of Russia’s central bank and the prospect that the country will default quickly on its obligations. Yet should Mr. Zelensky actually strike a deal with Mr. Putin, that could lead to a hard decision for the United States: whether to lift any of the sanctions that it has coordinated with nations around the world.

A Worse Alternative: Long, Slow Slog

Despite his military’s logistical problems, Mr. Putin appears intent on intensifying his campaign and laying siege to Kyiv, the capital; Kharkiv, the country’s second-largest city; and other Ukrainian urban centers.

But even as Mr. Putin presses on with his strategy to pound Kyiv into submission, Russian air and ground forces are confronting Ukrainians motivated to fight, senior Pentagon and U.S. intelligence officials said.

William J. Burns, the C.I.A. director, told lawmakers last week that he is anticipating an “ugly next few weeks.”

“I think Putin is angry and frustrated right now,” Mr. Burns said. He is likely to “try to grind down the Ukrainian military with no regard for civilian casualties,” he added.

Indeed, even as Russia widened its artillery, missile and bombing strikes on Sunday, Russian and Ukrainian forces were girding for what is shaping up to be a climactic battle in Kyiv.

Mr. Putin has demonstrated in past conflicts in Syria and Chechnya a willingness not only to bomb heavily populated areas but also to use civilian casualties as leverage against his enemies. Senior U.S. officials said the coming weeks could see a long, drawn-out fight with thousands of casualties on both sides, as well as among the roughly 1.5 million citizens remaining in the city.

Russian and Ukrainian forces are now pitted in fierce street fighting in the suburban towns around the capital, while Ukrainian troops ambush the Russians with Javelin anti-tank missiles supplied by NATO and the United States.

Russian forces greatly outnumber the Ukrainian army, and might grind them down.

Lt. Gen. Scott D. Berrier, the director of the Defense Intelligence Agency, told lawmakers last week there was a limit to how long Kyiv could hold on as Russian forces edged closer from the east, north and south, tightening the vise. “With supplies being cut off, it will become somewhat desperate in, I would say, 10 days to two weeks,” General Berrier said.

Another senior U.S. official, speaking on condition of anonymity to discuss confidential intelligence assessments, said it could take up to two weeks for Russian forces to encircle Kyiv and then at least another month to seize it. That would require a combination of relentless bombardment and what could be weeks or months of door-to-door street fighting.

“It will come at a very high price in Russian blood,” said retired Adm. James G. Stavridis, the former supreme allied commander for Europe. That high cost, he added, could cause Mr. Putin to destroy the city with an onslaught of missiles, artillery and bombs — “continuing a swath of war crimes unlike any we have seen in the 21st century.”

Abandoning Plan A, and Dividing the Nation

The Russian assault has so far failed to achieve any of Mr. Putin’s initial objectives. But on the battlefield, he is closer to some goals than others.

Beyond Kyiv, the northern cities of Kharkiv, Chernihiv and Sumy remain encircled, or nearly so, and continue to suffer heavy Russian shelling. Progress in the east and south, while slow, has been grindingly steady. But it also hints what a divided Ukraine might look like.

Russian forces are still subjecting Mariupol to siege and bombardment, but are close to securing that strategic southern port city and, with it, a land bridge from Crimea in the south to the Donbas region in the east that’s been controlled by Russian-backed separatists since 2014.

And if Russia can seize Odessa, a pivotal Black Sea port city, and perhaps the remaining Ukrainian coast to the southeast, it would deprive Ukraine of important access to the sea.

Senior Pentagon officials said the key issue now is maintaining extreme pressure on Russia in hopes that Mr. Putin will cut his losses and settle for the Russian-speaking south and east.

Yet the Russian attacks in western Ukraine over the past two days underscore Mr. Putin’s continued determination to control the entire country, starting with Kyiv. It remains unclear how he would find the forces to occupy it, which could require a bloody, yearslong guerrilla war.

“The most probable endgame, sadly, is a partition of Ukraine,” said Mr. Stavridis, pointing to the outcome of the Balkan wars in the 1990s as a model. “Putin would take the southeast of the country, and the ethnic Russians would gravitate there. The rest of the nation, overwhelmingly Ukrainian, would continue as a sovereign state.”

Worst-Case Scenario: Escalation

The fear now is that the war could expand.

The more the fighting moves west, the more likely it is that an errant missile lands in NATO territory, or the Russians take down a NATO aircraft.

Mr. Putin has used chemical weapons before against political opponents and defectors, and he might be inclined to do so again. Using battlefield nuclear weapons would cross a threshold, which most American officials believe even Mr. Putin would not do unless he believed he was facing the need to withdraw his troops. But the possibility of a nuclear detonation has been discussed more in the past two weeks than in years, officials say.

And finally, there are cyberattacks, which have been strangely missing from the conflict so far. They may be Mr. Putin’s most effective way of retaliating against the United States for grievous harm to the Russian economy.

So far there are none of the procedures in place that American and Russian pilots use over Syria, for example, to prevent accidental conflict. And Mr. Putin has twice issued thinly veiled reminders of his nuclear capabilities, reminding the world that if the conflict doesn’t go his way he has far larger, and far more fearsome, weapons to call into play.