The violence of Russia’s invasion of Ukraine hasn’t yet reached Odesa, the country’s third-largest city, but Vladimir Putin’s navy was headed there Wednesday night.

Several Russian warships from Crimea were headed to Odesa. An amphibious assault could come as soon as Thursday, U.S. officials told Fox News.

It comes as Russian forces attack the waterfront cities of Kherson and Mariupol.

FILE PHOTO: Russian Navy vessels are anchored in a bay of the Black Sea port of Sevastopol

Russian Navy vessels are anchored in a bay of the Black Sea port of Sevastopol in Crimea May 8, 2014.    (REUTERS/Stringer/File Photo)

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In Odesa, a city of roughly one million people on Ukraine’s southern coast, supermarkets ran out of supplies days ago, according to Kate, a 24-year-old local manager who asked to withhold her last name due to safety concerns amid a full-scale invasion of her country.

She shared pictures of empty supermarket shelves. Although she said she stocked up on food and water, she doesn’t see an opportunity to resupply.

"I went out to the store to replenish our supplies," she told Fox News Digital. "But the store shelves are empty. The store warehouses are just as empty. New supplies are now impossible."

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She’s spent the last week waiting for air raid sirens, running down to a fallout shelter when they blare. The city has a 6 p.m. curfew.

She lives with her cat and her boyfriend, who is medically ineligible to serve in Ukraine’s defense forces, but they’ve remained in Odesa, where she estimated only 20% of the population has stayed.

Russian attacks intensified Wednesday, she said, and she lost contact with friends in besieged Kharkiv and Kyiv earlier in the week.

"No one else got in touch," she said. "And there is an explanation: For days, there has been shelling are Kharkiv and Kyiv."

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Ukraine officials have accused Russian forces of targeting civilians with artillery and airstrikes in both cities.

In Kharkiv, a family of five burned to death in their car, according to the city’s mayor. Four other people ventured out of a bomb shelter for water and were killed by another bomb.

And in Kyiv, a Russian missile strike on a civilian TV tower killed at least five people near the Babyn Yar Holocaust memorial.

Militia members also told Fox News Digital Wednesday that a Russian rocket aimed at the Ukrainian army general’s staff was intercepted by Kyiv’s air defenses – sending a piece of debris crashing to the ground near the city’s crowded central train station. The area had been packed hours earlier with hundreds of women, children and elderly residents hoping to flee before an expected Russian assault.

"We had gunfire today," Kate said Wednesday evening. "There were five explosions, but so far, not on infrastructure. The siren went off about 10 times."