Sunday’s election in Ukraine could be the beginning of the end of the crisis that has enveloped the West’s relations with Russia over the past three months. Or, if Russian-inspired violence breaks out, it could be the start of far more serious and widespread international upheaval.

The United States and its European allies believe they have taken every possible step to ensure a free and fair presidential choice. By their calculations, peaceful voting should be possible in 200 of about 215 Ukrainian electoral districts. More than 1,000 international observers are in place.

“So far,” a senior European diplomat said this week, prospects look “surprisingly good.”

Russian President Vladi­mir Putin, who has called Ukraine’s current, ­Western-backed government illegitimate, appeared to blink Friday. Moscow will “respect the choice made by the people of Ukraine,” he said, and is prepared to “work with the new authorities after the election.”

Western governments express deep uncertainty at what Russia will do, and it was symptomatic of their equally deep mistrust of Putin that few took him at his word. U.S. officials parsed his language as leaving a hole big enough to drive a brigade of Russian soldiers through.

“Pull the rest of your troops back, then,” said State Department spokeswoman Marie Harf, repeating demands that Putin end the threatening deployment of Russian military ­forces along Ukraine’s eastern border. “Put your money where your words are. Come on.”

Russian Gen. Valery Gerasimov, chief of the general staff, said Friday that the troops have started to withdraw, a claim Putin has made several times in recent weeks.

The Pentagon agreed for the first time that there had been some “small-scale” troop movement away from Ukraine and evidence of preparations for more. But “tens of thousands of ready troops along that border . . . still are escalating the tension there in Ukraine, and we continue to call for the removal of all those troops,” said Rear Adm. John F. Kirby, the Pentagon’s press secretary.

Fighting continued in at least two areas of eastern Ukraine on Friday, where pro-Russian separatists — armed, financed and directed by Moscow, according to Washington and its allies — have occupied public buildings and declared an independent republic.

Last week, Secretary of State John F. Kerry said the United States and its principal European partners agreed that “if Russia or its proxies disrupt the election,” the allies would “impose sectoral economic sanctions as a result.”

The Obama administration has said it is “ready” with a new round of greatly increased punitive measures against major Russian economic sectors, including defense, energy, banking and mining. U.S. officials said that action, if warranted, could come as soon as late next week.

But on the eve of the vote, the allies offered no firm definition of what would constitute sanctionable disruption and whether it would have to be undertaken by the Russians themselves or their Ukrainian “proxies.”

What Russia says after the vote, in terms of fulfilling Putin’s pledge to work with a future government, is seen as equally important, even if there is no direct interference in the election itself.

Finding common ground in Europe on sanctions has been “extremely difficult,” said the senior diplomat, speaking on the condition of anonymity to discuss the situation frankly. While values unite the two sides of the Atlantic, he said, interests sometimes diverge. The United States does about $25 billion in annual trade with Russia; the European Union about $450 billion.

Former Soviet satellites, including the Baltic states and Poland, are eager for harsh measures. Some E.U. members, including Greece and other smaller states whose economies are still suffering the effects of near-bankruptcy, fear the fallout from a cutoff of Russian tourism and trade.

German Chancellor Angela Merkel, whose leadership is seen as decisive, said in a visit to Washington early this month that her government is committed to sanctions against major Russian economic sectors if necessary.

Despite extensive business lobbying against them, Germany’s leading industrial group said last week that it would go along with sanctions, albeit with a “heavy heart.”

Beyond the West’s reluctance to punish itself by punishing Russia, there is trans-Atlantic agreement on the eventual need to bring relations with Moscow back on an even footing.

“We have to deal with Russia in the future” beyond Ukraine’s elections, the European diplomat said. “There is concern that the Ukraine crisis does not contaminate future cooperation,” including on issues such as Syria and Iran. “So far, it has not.”

Putin made the same point Friday. “We still hope that common sense and the realization of their national interests will push all our partners in Europe and the U.S. into continuing their partnership with Russia,” he said at a conference in St. Petersburg, according to the Russian news agency Interfax.

“Despite our varying, maybe diametrically non-overlapping approaches in assessing critical situations, we nevertheless continue cooperation,” Putin said. The United States suspended military cooperation with Russia over Ukraine, he noted, “but they have not suspended military cargo transit to and from Afghanistan via our territory, because it is convenient for them. As a matter of fact, we have not refused it, either.”

Russia, he added, is still participating with the United States and the Europeans in nuclear negotiations with Iran. U.S. officials agreed that there has been no change in Russia’s cooperation in the talks.

Western officials insist they only want a peaceful election in Ukraine and have no preference among the presidential candidates. But as they contemplate a future relationship with Russia, there is little doubt that they consider Petro Poroshenko, the wealthy businessman and former cabinet minister who leads the polls, as someone who could solve their problems in both Ukraine and Russia.

“We cannot invent the ideal leader,” the diplomat said, “and Ukraine has a history of very problematic leadership,” including massive corruption and rule by Russian-linked oligarchs.

But Poroshenko, he said, has a number of “useful qualities,” including “very good channels” to Moscow.