KYIV, Ukraine—Ukraine began a counteroffensive toward the eastern city of Izyum aimed at disrupting Russian supply lines into the Donbas region, officials said, as Ukrainian forces continued clearing villages north of Kharkiv and Russian President Vladimir Putin warned his Finnish counterpart that joining NATO would risk damaging relations with Moscow.

According to the Kremlin, Mr. Putin told Finland’s President Sauli Niinistö in a phone call Saturday that ending its decadeslong nonaligned defense policy by joining the North Atlantic Treaty Organization would be a mistake for Helsinki. The Finnish leader initiated the call to explain to Mr. Putin how his invasion of Ukraine had altered the security environment, prompting Finland to seek NATO membership in the coming days.

“The conversation was direct and straightforward and was conducted without aggravations. Avoiding tension was considered important,” Mr. Niinistö said.

Russian Deputy Foreign Minister Alexander Grushko, meanwhile, said that Moscow would need to take “adequate precautionary measures” if NATO were to deploy infrastructure for nuclear weapons near Russia’s borders, including in Finland.

A Ukrainian soldier takes part in a reconnaissance mission in a recently retaken village on the outskirts of Kharkiv, Ukraine.

Photo: Bernat Armangue/Associated Press

On Saturday, Russia’s Ambassador to the U.S., Anatoly Antonov, said the war in Ukraine is increasingly drawing Washington into conflict with Moscow.

“The situation today, is extremely, extremely dangerous. The U.S. is being drawn deeper and deeper into conflict with the most unpredictable consequences for relations between the two nuclear powers,” Mr. Antonov said on Russian television.

As the war entered its 80th day, Russian offensive operations in Donbas remained largely stalled following the failure of Russia’s ambitious attempt to cross the Siverskyi Donets river and encircle the metropolitan area of Severodonetsk, the capital of the Ukrainian-administered Luhansk region.

With Western weapons continuing to flow into Ukraine, Ukrainian officials are beginning to say that a pivot in the war might be near, with Kyiv switching from defense to offense to reclaim large parts of southern and eastern Ukraine that remain under Russian rule.

“A strategic break in Ukraine’s favor is under way. This process will take time. But, in the long term, these trends make Russia’s defeat inevitable,” Ukrainian Defense Minister Oleksii Reznikov

said in an address to Ukrainian citizens.

After Russia’s initial plans to seize the capital, Kyiv, failed amid Ukrainian resistance, Mr. Putin in late March ordered his forces to pull back from northern Ukraine and concentrate on seizing the entirety of the Donetsk and Luhansk regions that make up Donbas. Mr. Putin in February recognized the independence of the Moscow-created proxy states in Donbas, the Donetsk and Luhansk people’s republics, which controlled roughly one-third of these two regions at the time.

Russia has established the forward headquarters of its operations to conquer Donbas in the town of Izyum, which straddles the Siverskyi Donets river in the Kharkiv region. Ukrainian troops have begun to push successfully toward the town, the head of the Kharkiv regional military administration, Oleh Synehubov, said Saturday.

“The Izyum direction remains our hottest point. That’s where our armed forces have begun a counteroffensive,” he said in a video address. “The enemy is retreating in some directions, which is the result of the character of our armed forces.”

Workers remove documents from the ruins of Kharkiv’s regional administration building in Ukraine.

Photo: John Moore/Getty Images

Local residents receive humanitarian aid in Kharkiv, Ukraine.

Photo: John Moore/Getty Images

With Russia’s monthlong offensive in Donbas showing only limited results, a bold attempt to cross the Siverskyi Donets and encircle Severodonetsk that Russian forces began ahead of Victory Day on May 9 was meant to achieve a breakthrough. Instead, the failed crossing near the village of Bilohorivka has turned into a disaster for Russia, significantly slowing its momentum in Donbas.

The full scale of this Russian setback is emerging only now, with satellite imagery showing more than 70 Russian tanks, armored personnel carriers and other armor destroyed after Ukrainian artillery and airstrikes sank three pontoon bridges and shelled the Russian beachhead in Bilohorivka.

“We have never seen such dumb stubbornness, going with a frontal assault and trying to build pontoons in the same place three times in a row. But they still keep trying,” said Luhansk Gov. Serhiy Haidai.

He added that Ukrainian artillery keeps shelling the area and, according to intelligence intercepts, an entire Russian battalion is refusing orders to attempt yet another crossing in Bilohorivka. That claim couldn’t be independently confirmed.

While Moscow hasn’t acknowledged the events in Bilohorivka, accounts from Russian military officers and observers on Telegram have described it as one of the Russian military’s most catastrophic defeats in this war, calling for the dismissal and punishment of generals who devised the failed operation.

Reverse Side of the Medal, a channel close to the Wagner private military contractor that is actively involved in the war in Ukraine, pointed out sardonically that the Russian commanders in the Bilohorivka operation were “guided by the principle that the shell doesn’t fall into the same place twice and that if you don’t see the enemy, he cannot see you.”

Ukrainian artillery managed to destroy at least a battalion’s worth of Russian armor because it is employing drones and sophisticated reconnaissance technology to achieve precision, Wagner’s channel added. “The Armed Forces of Ukraine use the Western system of smart battle management, and we use a ruler on a paper map,” it said.

Relatives and friends attend the funeral in Lviv, Ukraine, for a serviceman who was killed in the Kharkiv region.

Photo: Emilio Morenatti/Associated Press

In Kyiv, Senate Minority Leader Mitch McConnell (R., Ky.) and GOP Sens. Susan Collins of Maine, John Cornyn of Texas and John Barrasso of Wyoming met Ukrainian President Volodymyr Zelensky on Saturday, according to Mr. Zelensky and a U.S. official. The meeting comes after House Speaker Nancy Pelosi (D., Calif.) and a delegation of Democratic lawmakers went to the capital city to meet with Mr. Zelensky earlier this month.

In a statement with accompanying video, Mr. Zelensky said the visit was a powerful signal of bipartisan support for Ukraine from Congress and the American people. A $40 billion aid package for Ukraine is stalled in the Senate over objections from GOP Sen. Rand Paul of Kentucky. Mr. McConnell and Senate Majority Leader Chuck Schumer (D., N.Y.) had pushed for the bill’s passage together Thursday to request unanimous agreement from all 100 senators to allow a vote on the bill immediately.

A U.S. official confirmed the visit and said it wasn’t publicized in advance because of security concerns. Representatives for the senators didn’t immediately respond to requests for comment.

North of Donbas, a string of Ukrainian military victories in recent days pushed Russian forces outside of field artillery range of the city of Kharkiv, where more than 2,000 residential apartment buildings have been destroyed in more than two months of pounding. In a sign of relative normalcy returning to Kharkiv, the municipality said public-transport services would resume Monday. It will initially be free of charge given that so many city residents have lost their jobs because of the war, said Mayor Ihor Terekhov.

Ukraine’s military has confirmed that its forces have reached the town of Ternova, on the border with Russia north of Kharkiv. “The enemy didn’t conduct active combat actions in the Kharkiv direction,” Ukraine’s General Staff said Saturday. “Its main effort was focused on pulling back troops from the city of Kharkiv, maintaining positions and protecting supply lines.”

The retreat of Russian forces from areas north of Kharkiv is covered by poorly armed recruits drafted in the Luhansk People’s Republic. In a video released on Telegram on Friday, these fighters said that their battalion, which fled to the Russian border north of Kharkiv, was stranded at the gate, with Russian authorities refusing to let them cross and threatening to imprison them if they don’t turn around and fight.

—Eliza Collins, Lindsay Wise, Dan Michaels and Vivian Salama contributed to this article.

Corrections & Amplifications
Russian troops made a failed attempt to cross the Siverskyi Donets river and encircle the metropolitan area of Severodonetsk. An earlier version of this article misspelled it as Siverskiy Donets. (Corrected on May 14)

Write to Yaroslav Trofimov at yaroslav.trofimov@wsj.com and Mauro Orru at mauro.orru@wsj.com