As Russian forces battle for control of a key eastern Ukrainian city, up to 12,000 civilians may be trapped in the crossfire, according to the Norwegian Refugee Council (NRC).

"I am horrified to see Severodonetsk, the thriving city where we had our operational headquarters, become the epicentre of yet another chapter of the brutal war in Ukraine," NRC Secretary-General Jan Egeland said in a statement Tuesday. "We fear that up to 12,000 civilians remain caught in crossfire in the city, without sufficient access to water, food, medicine or electricity. The near-constant bombardment is forcing civilians to seek refuge in bomb shelters and basements, with only few precious opportunities for those trying to escape."

Over the past week, the Oslo-based humanitarian organization has been working with local Ukrainian partners to provide thousands of monthly food and hygiene parcels to civilians remaining in Severodonetsk and the greater Luhansk Oblast, according to Egeland. The city is the last still held by Ukrainian forces in Luhansk Oblast.

"But now the intensified fighting makes aid delivery impossible," he added. "We cannot save lives under the hail of grenades."

NRC has been operational in Ukraine since 2014, serving people affected by conflict in the Luhansk and Donetsk oblasts of eastern Ukraine's war-torn Donbas region.

"Almost one hundred days since the war in Ukraine escalated, we have seen bombs destroy critical infrastructure across the country and reduce entire cities like Severodonetsk to rubble," Egeland said. "More than 14 million men, women and children are displaced within Ukraine or sheltering in other countries with no idea when they will be able to safely return to their homes."

Russian and Ukrainian forces are believed to be fighting street by street on the outskirts of Severodonetsk, a key city in Ukraine's east, the U.K. Ministry of Defense said Tuesday in an intelligence update.

"Russia's capture of Lyman supports its operational main effort, which likely remains the encirclement of Sieverodonetsk and the closure of the pocket around Ukrainian forces in Luhansk Oblast," the ministry said. "Heavy shelling continues, while street fighting is likely taking place on the outskirts of Sieverodonetsk town."

After several days of fighting, the Russian military claimed Saturday to have fully seized the strategic town of Lyman, which serves as a railway hub in the Donetsk Oblast, west of Severodonetsk.

"Russia's political goal is likely to occupy the full territory of Donetsk and Luhansk Oblasts," the ministry added. "To achieve this, Russia will need to secure further challenging operational objectives beyond Sieverodonetsk, including the key city of Kramatorsk and the M04 Dnipro-Donetsk main road."

The European Union has agreed to partially ban Russian oil exports to the EU, leaders announced on Monday.

The new ban "immediately" covers more than two-thirds of oil imports from Russia, "cutting a huge source of financing for its war machine," Charles Michel, president of the European Council, tweeted on Monday.

"We agreed in Versailles to phase out our dependency on Russian gas, oil & coal as soon as possible," European Commission President Ursula von Der Leyen tweeted. "But this requires a plan."

-ABC News' Christine Theodorou

Bridget Brink, the new U.S. ambassador to Ukraine, has arrived in Kyiv, the U.S. Embassy confirmed.

Brink was nominated by President Joe Biden in April and confirmed by the Senate on May 18.

-ABC News' Christine Theodorou

White House national security advisor Jake Sullivan hinted Thursday of more sanctions coming against Russia in the "next week or two" aimed at targeting ways Moscow is evading sanctions already imposed.

“Where our focus will be over the course of the coming days is on evasion,” Sullivan said Thursday at the Economic Club of Washington. “As Russia tries to adjust to the fact that it’s under this massive economic pressure, what steps do they take to try to evade our sanctions and how do we crack down on that? And I think we'll have some announcements in the next week or two that identify targets that are trying to facilitate that evasion both inside Russia and beyond."

When Sullivan was asked whether sanctions will automatically be lifted if a negotiated peace deal between Russia and Ukraine is worked out, he appeared cautious with his words, saying, “a lot of that depends on what the shape and scope” of the agreement is.

“A lot of it depends on what the Ukrainians, in consultation with us and the Europeans come to agree to," Sullivan said. "You know, we're not going to do a deal over the head of the Ukrainians where we give a bunch of sanctions relief to Russia. But if some measure of sanctions relief were built in to some credible diplomatic solution led by the Ukrainians, that's something that we would happily discuss."

But Sullivan said Russian oligarchs shouldn't expect to ever get back their yachts and other assets seized under sanctions that have been imposed, saying the ultimate goal is "not to give them back” once the war is over.

“The president is actively looking at how we can deal with the fact that as we seize these assets, our goal is not to give them back. Our goal is to put them to a better use than that," Sullivan said. "But I'll be careful in what I say today because there's an ongoing kind of policy process around how we end up dealing with that question. But, rest assured, that the goal is not just to sit on them for a while."

-ABC News' Justin Gomez