With the push for secession now stalling, anti-Russian sentiment on the rise in southern and eastern Ukraine, and Ukrainian patriotism surging throughout most of the country, Mr. Putin has begun to distance himself from the separatists. He has not officially acknowledged the results of the May 11 referendums, and he has suggested that Sunday’s presidential election could help stabilize Ukraine. His ultimate objective remains murky, but for now he seems to be backing off.

Several government ministers and political leaders I met in Kiev over the past week expressed cautious optimism at the pushback from the Ukrainian people. “I think this will end up all right,” one minister said.

These officials and Western diplomats said the Russian government was about to significantly cut the price of the gas it sells to Ukraine. A back-channel dialogue between Moscow and Kiev has begun. On May 17, Prime Minister Arseniy Yatsenyuk of Ukraine told me and a small group of Western writers and academics that he had recently met a leading envoy from the Kremlin and believed Mr. Putin was looking for an “exit ramp” away from violent conflict.

The way out may involve some of Ukraine’s oligarchs, who want more trade with Europe and North America, but also have significant business interests in the Russian market. Rinat Akhmetov — the richest man in Ukraine, a onetime ally of the deposed President Viktor Yanukovych and a mainstay of the political scene in the Donbass region — has taken a stand against the separatists. This week he denounced the agents of the Donetsk People’s Republic as armed thugs. He has organized the steelworkers and coal miners in his employ into civilian patrols throughout the Donetsk region. He has called for strikes to protest any further lawlessness. Such positioning by the oligarchs will be crucial to the stability of the country.

The front-runner in Sunday’s election is Petro Poroshenko, who made a fortune in the confectionery and automobile-manufacturing businesses. Mr. Poroshenko is pro-E.U. and was on the front lines of the push to oust Mr. Yanukovych last winter, but he is also well known to Russia’s elite as a businessman and a pragmatist. This combination would make him a credible interlocutor for the Kremlin, especially now that Mr. Putin’s expansionist ambitions have crashed against the determination of Ukrainians, making way for a new dynamic between the two countries.

Adrian Karatnycky is a senior fellow at the Atlantic Council, where he co-directs the Ukraine in Europe program.