The capture of the town, 5km from the border, would be Russia's most significant gain since it launched an incursion into the region on Friday, opening a new front in its invasion and forcing Ukraine to rush in reinforcements.
The assault keeps Ukraine's forces, also holding the 1000km front line in east and south, off balance ahead of what Zelenskiy has said could be a big Russian offensive in the coming weeks.
Russia has been slowly making ground in the east for months.
"The situation is extremely difficult. The enemy is taking positions on the streets of the town of Vovchansk," Oleksiy Kharkivskyi, Vovchansk's patrol police chief, said on Facebook.
Dmytro Lazutkin, a spokesman for the defence ministry, said "some" Russian infantry groups had entered the town.
Ukrainian troops later managed to "partially" push them back, the general staff said but "defensive actions" raged in the north and northwestern outskirts.
Russia's defence ministry said on Wednesday its forces captured two more settlements in the region, bringing the total count to 12 since Friday.
Every day, Russian terrorists use missiles and guided aerial bombs to strike Kharkiv and other cities and communities in the border and frontline areas.— Volodymyr Zelenskyy / Володимир Зеленський (@ZelenskyyUa) To increase protection against Russian terror, our air defense must be strengthened, including by expanding the fighter jet… pic.twitter.com/LOxJ00KCVCMay 14, 2024
The latest were Hlyboke and Lukyantsi, it said, both about 25km from Kharkiv's outskirts.
Ukraine said late on Tuesday it pulled back to new positions in the Vovchansk and Lukyantsi areas due to "a consequence of enemy fire and storming action".
Police remained in Vovchansk and were continuing to help people leave, Kharkivskyi said.
Nearly 8000 people have been relocated from Vovchansk and border areas since Friday's assault.
Zelenskiy has postponed all his foreign travel planned for the coming days, his spokesman Sergiy Nykyforov said, after the Ukrainian leader held a daily conference call with senior military figures to discuss the situation in Kharkiv region and the supply of weapons.
Ukraine is trying to snuff out the assault in the Kharkiv region, while holding the line against Russia's main thrust in the eastern Donbas region and guarding against potential new border incursions.
The top military spy has warned that Russia had small groups of forces located to the north of Kharkiv region along its border with the Sumy region.
Ukraine's shortage of troops is compounded by months of delayed weapons deliveries, in particular from the United States where Congress took six months to approve a major aid package.
"For Russians, now is actually a window of opportunity ... The Russians feel it, they have accumulated enough resources," Serhii Rakhmanin, MP and a member of the Ukrainian parliamentary committee for security and defence, told Reuters.
He expected the next three months to be the most critical for Ukraine but anticipated that the situation would improve due to fresh weapons supplies, if they arrive in time.
The deteriorating situation in Kharkiv region coincided with a visit to Kyiv by Secretary of State Antony Blinken, who said the US was aiming to ensure the speedy delivery of much needed weapons to battlefield.
"We're rushing ammunition, armoured vehicles, missiles, air defences - rushing them to get to the front lines to protect soldiers, to protect civilians," he said.
Ukraine says the Russian assault into the northeast does not present an imminent threat to the region's city of Kharkiv, Ukraine's second largest, that is home to 1.3 million people.
