A pivotal showdown approaches in eastern Ukraine as more villages fall to Russia.

Approaching a pivotal moment in their invasion of Ukraine, Russian forces have tightened their vise around two key eastern cities, raising the risk their slow, brutal advance will capture the cities and trap the Ukrainian troops defending them.

The fall of the two neighboring cities, Sievierodonetsk and Lysychansk, would all but complete Russia’s conquest of Luhansk Province, a major part of the Donbas region that the Russians are attempting to seize in the four-month-old war. That would give a strategic and symbolic victory to President Vladimir V. Putin, and open avenues for Russia’s military to advance deeper into Ukraine.

The Russians have captured three more villages south of the cities, moving them within easier artillery range of Lysychansk, where Ukrainian forces are digging in on high ground for what could be a pitched battle for the city. Moscow’s forces already control most of Sievierodonetsk, to the east, which sits on lower ground and has been reduced to ruins by the Russian bombardment.

“Russian forces are getting closer to Lysychansk,” Serhiy Haidai, the Ukrainian regional administrator, warned Wednesday on Telegram, the messaging app, as he confirmed the capture of the villages Mirna Dolina, Pidlisne and Toshkivka.

After failing to seize Ukraine’s two biggest cities, Kyiv, the capital, and Kharkiv, early in the war, Mr. Putin shifted his focus to Donbas, the eastern region that is Ukraine’s industrial heartland, where Russian-backed separatists have waged war since 2014. The invasion has been much harder for Russian forces, who have suffered setbacks and heavy losses — as have the Ukrainians — and Mr. Putin is seen by Western analysts as being eager for something he can call success.

For Ukrainians, the main concern in the short run is Russian artillery moving close enough to make it harder for them to use a highway that runs southwest from Lysychansk and is the main supply line and evacuation route for Ukrainian forces and civilians in the city, which has been mostly emptied of its prewar population of about 100,000.

That enhances the possibility that the Ukrainian fighters defending Lysychansk could be encircled and cut off. Ukrainian fighters in the port of Mariupol on the Sea of Azov were encircled for weeks before surrendering in early May.

The battle around Sievierodonetsk, which has raged for weeks, grew more dire for the Ukrainians on Sunday, when Russian troops broke through a key defensive position at Toshkivka, prompting Ukrainian forces to rush reinforcements to the area.

Ukrainian forces still control all of Lysychansk, but in Sievierodonetsk, across the Siversky Donets River, they hold only a chemical plant where civilians are also reported to be sheltering, Mr. Haidai said on Monday.

In addition to Luhansk province, the Russians control about half of Donetsk province and are pushing from the east, north and south to take more territory. But analysts say that Russia’s battered forces face an even more difficult battle to seize more of Donetsk.

A British military intelligence report on Wednesday said that Moscow-backed separatist forces in Donetsk had reported the killing of more than 2,000 of its personnel this year and the wounding of almost 9,000. Those figures amounted to 55 percent of its original force, the British assessment said, which “highlights the extraordinary attrition Russian and pro-Russian forces are suffering in the Donbas.”

Ukrainian forces have also taken heavy casualties and are relying more frequently on undertrained units to hold parts of the eastern front line.

For both Russia and Ukraine, “the ability to generate and deploy reserve units to the front is likely becoming increasingly critical to the outcome of the war.”

President Volodymyr Zelensky of Ukraine said in an overnight address that his country’s army, with the help of “tactical moves,” was strengthening its defense in Luhansk, a possible reference to the arrival of longer-range artillery supplied by the United States and some European governments. But he acknowledged that Luhansk was “really the toughest area right now.”