The fight underscores the central role of artillery in the east of Ukraine as Russia tries to avoid losing more manpower in urban fighting. The city’s fall would likely free up Russian soldiers for their next assault on the Ukrainian cities of Slovyansk and Kramatorsk, which would be the last redoubts of Ukrainian forces in the east.

Serhiy Haidai,

the governor of Luhansk region, which together with neighboring Donetsk makes up the Donbas region, said Monday morning that all critical infrastructure in Severodonetsk had been destroyed, as had 60% of the residential housing. Speaking on Ukrainian television, he described the pattern of the battle: several hours of Russian shelling, followed by a push by troops to advance further into the city.

“The smell of corpses is constant, because they do not take away the bodies,” Mr. Haidai said. “They are firing everything in the arsenal of the Russian Federation.” Ukrainian troops were trying to hold out until more heavy weapons could be delivered, he said.

On Monday afternoon, Mr. Haidai said that officials were halting evacuations after an armored evacuation vehicle came under fire, killing a French journalist.

The vehicle had been on its way to pick up 10 people when shrapnel pierced the armor of the car and fatally struck the journalist, who was reporting on evacuation efforts, Mr. Haidai said on his Telegram channel. He didn’t identify the journalist.

Russian Foreign Minister

Sergei Lavrov

told French television channel TF1 that what he called the liberation of the two regions of Donbas, which Moscow considers independent states, is “an unconditional priority.”

Russia’s forces were so concentrated on Severodonetsk that they likely wouldn’t be able to conduct major operations elsewhere in the country, according to the Institute for the Study of War. Still, Russia continued airstrikes elsewhere in Ukraine, an indication that Moscow’s ambitions stretch beyond Donbas.

The Sumy and Kharkiv regions, in northern Ukraine, and Kryvyi Rih in the south, were all hit with airstrikes, according to Ukrainian officials Monday morning. In addition, officials said reinforcements were being sent to Russian positions around Ukraine.

Russia said it had also struck targets in the country’s south, including Ukraine’s military southern command.

Kirill Stremousov, the Russian-appointed deputy head of the military-civil administration of the Kherson region, said farmers in the region had begun exporting wheat to Russia, Russian newswires reported. Ukraine has previously accused Russia of stealing its grain supplies, as well as blocking grain exports, which has led to food shortages in some parts of the world.

Ukrainian forces in the south, near Kherson, conducted a successful counteroffensive over the weekend, according to the Institute for the Study of War. Though they were unlikely to retake more territory, they might force Russia to deploy more resources to the region.

The governor of Russia’s Kursk region said Monday that the Defense Ministry would be sending more forces to the region to protect against alleged Ukrainian cross-border attacks. Russia has blamed Kyiv for a number of strikes on fuel depots and critical infrastructure inside Russian territory.

President Biden, arriving back at the White House Monday morning from Delaware, was asked whether the U.S. was planning to send long-range rocket systems to Ukraine.

“We are not going to send to Ukraine rocket systems that strike into Russia,” Mr. Biden said, without elaborating.

The Wall Street Journal on Friday reported that the Biden administration is expected to announce as early as this week it will send to Ukraine long-range rocket systems Kyiv says are necessary to fight off the Russian onslaught in the Donbas region.

Among the weapons the U.S. is expected to provide are multiple launch rocket systems, or MLRS, which are mobile artillery launchers that can fire dozens of miles farther than any current system in Ukraine’s control.

After the president’s remarks Monday, an administration official said that sending MLRS to Ukraine is under consideration for battlefield use, but that “nothing is on the table with long-range strike capabilities” that could hit targets in Russia.

Mr. Biden later invoked Ukraine in a Memorial Day speech to families of slain U.S. service members at Arlington National Cemetery.

The war Ukrainians are waging is “part of a larger fight that unites all people,” Mr. Biden said. “It is a fight that so many of the patriots whose eternal rest is here on these hallowed grounds were part of—a battle between democracy and autocracy, between liberty and repression.”

Meanwhile, leaders of the European Union’s 27 countries began meeting in Brussels on Monday afternoon to discuss issues related to Russia’s invasion of Ukraine.

EU leaders have been trying for weeks to agree on a sixth package of sanctions on Russia, this one focused on oil exports. Some EU countries that depend heavily on Russian oil, particularly Hungary, have been balking at cutting off the flow, arguing the economic impact would be too great.

On Monday, EU officials representing member states agreed on a preliminary text for leaders to review, but it remained unclear if Hungarian Prime Minister

Viktor Orban

would accept the draft, according to people familiar with the discussions.

The draft text, potentially for adoption at the summit, says leaders agree to a sixth package of sanctions, covering crude oil and petroleum products from Russia “with a temporary exception for crude oil delivered by pipeline,” according to a draft reviewed by the Journal.

Landlocked Hungary receives all its Russian oil by pipeline. Piped oil accounts for about one-third of EU imports of Russian oil, with the rest coming by ship.

Ukrainian President

Volodymyr Zelensky

will address the gathering by video and is likely to urge both greater support for his country and intensified efforts to punish Russia and its president,

Vladimir Putin.

In a video address released late Sunday night, Mr. Zelensky reiterated the country’s need for more weapons to push back the Russians. He also contrasted the brutal fighting in the east with the relative peace in the capital, as Kyiv residents celebrated the 1,540th anniversary of the city’s founding.

On Sunday afternoon, Kyiv residents packed the capital’s streets, sat by the riverfront and watched outdoor performances of traditional Ukrainian song and dance.

“It is for such happiness to live, live your normal life and just quietly walk the streets of your own city that this war is going on now. That’s what we are fighting for in it,” Mr. Zelensky said. “Freedom and peace is what could really be felt in Kyiv today. Even in spite of the air-raid siren that sounded today as well.”

—Alex Leary, Daniel Michaels, Illia Oliinyk and Laurence Norman contributed to this article.

Write to Ian Lovett at ian.lovett@wsj.com and Thomas Grove at thomas.grove@wsj.com