Ukrainian aircraft launched three strikes on areas of concentration of Russian forces, the Ukrainian military said on Tuesday night.
Bakhmut had a population of about 70,000 before the war but has been ruined during months of fighting as a focal point of Russian assaults and determined Ukrainian defence.
"The most difficult part, as before, is Bakhmut and the fighting that is essential for the city's defence," Ukrainian President Volodymyr Zelenskiy said in his nightly video address.
"Russia in general takes no account of people and sends them in constant waves against our positions, the intensity of the fighting is only increasing," Zelenskiy said.
A Russian takeover of Bakhmut would open the way to seizing the last remaining urban centres in the industrial Donetsk province.
Although most of the Russian attacks were focused on Bakhmut and other towns and villages in Donetsk, the Ukrainian military said Russian forces shelled more than 20 settlements in the northern regions of Chernihiv, Sumy and Kharkiv.
Reuters was not able to verify battlefield reports.
Ukrainian military analyst Oleh Zhdanov said Russian forces had driven a wedge between two villages north of Bakhmut, Berkhivka and Yahidne, in their bid to surround the city.
"This breakthrough on Bakhmut's northern flank poses a clear threat to us," he said in comments posted on social media.
In Washington, senior US defence official Colin Kahl told a congressional hearing that the front lines of the war were a "grinding slog" and there was nothing to suggest "the Russians can sweep across Ukraine and make significant territorial gains anytime in the next year or so".
Kahl spoke during a hearing focused on oversight of the nearly $US32 billion ($A47 billion) in military aid President Joe Biden's administration has provided to Ukraine since Russia's invasion on February 24 last year, including drones, long-range artillery systems, and air defence capabilities.
Ukraine has sought weaponry to protect itself from waves of Russian missile and drone attacks that in the depths of winter damaged the power grid and other infrastructure, killed hundreds of civilians and left millions with no electricity or water.
As part of an investigation into whether the attacks contravened the Geneva conventions on military conflict, the International Criminal Court's top prosecutor, Karim Khan was in Ukraine on Tuesday.
"Generally we see clearly a pattern, I think, in terms of the number, scale and breadth of attacks against the power grids of Ukraine and we need to look at why that's taking place; are they legitimate targets or not?" Khan told reporters in the town of Vyshhorod just north of the capital Kyiv.
Russia says its attacks are legitimate strikes aimed at weakening the enemy's military but Ukraine casts them as a means of intimidating the public.
Elsewhere, foreign ministers from around the world will meet in New Delhi on Wednesday and Thursday in the shadow of Russia's war in Ukraine and spiralling US-China tensions.
The meeting will be attended by Russian Foreign Minister Sergei Lavrov and US Secretary of State Antony Blinken while China is also expected to send its foreign minister, Qin Gang.
India does not want Ukraine to dominate the event, but it will top the agenda, said an Indian foreign ministry official.
Kremlin spokesman Dmitry Peskov on Tuesday repeated Moscow's stance that it is open to peace negotiations but Ukraine and its Western allies must accept Russia's annexation of Donetsk, Luhansk, Kherson and Zaporizhzhia regions after referendums last September that most governments said were illegal.
Despite several battlefield setbacks in what Russia describes as a "special military operation" to protect its security interests, its forces still control about a fifth of territory in its European-leaning neighbour Ukraine.
Zelenskiy's government has ruled out talks with Russia and has demanded that its troops withdraw to Ukraine's 1991 borders, the year the Soviet Union collapsed.
