Russia-Ukraine updates: US sanctions Russian military shipbuilder, diamond miner

One of Russia’s negotiators in cease-fire talks with Ukraine said he believes "substantial progress" has been made and that the two sides might reach a "unified position" in the near future, the Russian state news agency TASS reported Sunday.

Leonid Slutsky, a senior Russian lawmaker who has taken part in the talks, was quoted by TASS as saying, “The progress in the talks between Russia and Ukraine in the near future can grow into a unified position of both delegations."

The two sides have held three rounds of face-to-face talks in Belarus and have been talking by video conference in recent days.

Russia has suggested its conditions for ending the war would require Ukraine to change its constitution to guarantee it will never join a political bloc, specifically NATO or the European Union, as well as recognizing Crimea as part of Russia and the two Russian-controlled separatist areas in the Donbas region of in eastern Ukraine as independent.

Ukraine previously said Russia continues to demand its surrender.

Ukraine's lead negotiator and presidential adviser Mykhailo Podolyak also issued a hopeful statement Sunday, saying the two sides are moving toward compromises and that they are now discussing concrete points. Podolyak told the independent Russian newspaper Kommersant that Russia is looking at the situation “far more properly."

"They have started to talk about something and not just throw out ultimatums," Podolyak told the newspaper. But he said more time was needed for Russia to fully understand the reality of its situation and the need to further compromise on its demands.

According to Kommersant, Podolyak said working groups are developing the legal aspects of documents the two sides might eventually be able to sign. For now, both have agreed to keep the proposals confidential until they have an agreement, he said.

He said the negotiations have focused on conditions for a cease-fire and peace agreement, how to compensate Ukraine for the damage to its infrastructure and the process of withdrawing Russian troops. Notably, he said the key point of a peace agreement is the issue of "security guarantees" for both Russia and Ukraine.

“We have all the proposals, which one way or the other protect these interests, get Ukraine out from under attack from the point of view that similar situation won’t be repeated, get Ukraine to some kind of compensatory things in the right sense. And I emphasize again -- the Russian side already looks at [the situation] far more properly. But some time must pass still so they understood 100% the situation in which very much Russia, not Ukraine, has fallen," Podolyak said.

Podolyak said he hopes a fourth round of talks will be held in person.

A Russian airstrike killed 35 people early Sunday morning at a military facility in western Ukraine, miles from the Polish border.

The Lviv Regional Military Administration said at least eight Russian rockets struck a training facility near Yavoriv, a town about 35 miles west of Lviv and 10 miles from the Polish border. The attack could be heard from Lviv.

Thirty-five people were killed and 134 were wounded, according to Maksym Kozytskyi, head of the regional military administration. It was unclear whether civilians were among the injured.

"Today the Lviv region was hit by missiles, preliminary by jets, stationed in Saratov, Russia, not flying into Ukrainian territory," Kozytskyi said. "Preliminary more than 30 missiles were launched. Anti-missile defense worked, some amount of missiles were shot down. The Yavoriv firing ground came under attack."

Yavoriv sits along one of the main routes used to bring supplies into Ukraine from Poland, a pathway also used in recent weeks by refugees fleeing the Russian invasion.

The facility struck on Sunday is near the International Center for Peacekeeping and Security, known as the base where for years NATO military instructors, including Americans, trained Ukrainian troops to fight against Russia in the east.

The administration said earlier on Sunday that nine people were killed and 57 were wounded.

The mayor of Ivano Frankivsk, a southwestern city, also confirmed a strike on the city’s airport early on Sunday. It’s the third such strike, official said.

-ABC News’ Kirit Radia, Julia Drozd, Patrick Reevell and Clark Bentson

Ukrainian Deputy Prime Minister Iryna Vereshchuk posted an update Saturday on the country's evacuation progress.

About 13,000 Ukrainians were evacuated Saturday, which was half the number from Friday, she said.

Nine out of 14 humanitarian corridors were open Saturday, according to Vereshchuk.

About 8,000 refugees left Sumy, more than 3,000 left Konotop, 800 people were taken out of Gostomel and about a thousand people evacuated Nemeshaevo, Vereshchuk said.

About a thousand people were rescued from Bucha, including patients at a nursing home, she said.

However, Russia didn't let a convoy through the checkpoint in Vasilyevka and as a result no one could evacuate from the Zaporіzhya region, Vereshchuk said.

A Ukrainian convoy also did not reach Mariupol, because it got stuck at a Russian checkpoint, according to Vereshchuk. The convoy will try to take people out from the locations Sunday, she said.

-ABC News' Jason Volack

A bipartisan group of U.S. senators announced Saturday that they are traveling to Poland to meet with Polish officials and visit refugee sites to reaffirm the United States' commitment to Poland, Ukraine and other allies in response to Russian President Vladimir Putin’s invasion of Ukraine.

The group is made up of Sens. Rob Portman, R-Ohio, Amy Klobuchar, D-Minn., Roger Wicker, R-Miss., and Richard Blumenthal, D-Conn.

The group was also expected to meet with the U.S. 82nd Airborne Division stationed in Poland on the trip this weekend.

"The Polish people continue their unwavering support for NATO after 23 years of fighting alongside the U.S. and our allies. This bipartisan delegation will send a clear message that the U.S. is thankful for their support of Ukraine and their offer of providing fighter jets, as the people of Ukraine continues to endure brutal Russian atrocities," Portman said in a statement.

Echoing U.S. support for Ukraine, Blumenthal said, "As Putin continues his brutal, savage invasion, I’m traveling with a bipartisan group of Senators to the Ukrainian border in Poland to see first-hand the heartbreaking, exploding humanitarian crisis — and learn how America can magnify its military assistance to brave Ukrainian freedom fighters."

-ABC News' Trish Turner