The Russian troops, including mercenary fighters from the Wagner Group, are trying to cut the Ukrainian defenders' supply lines to the city, scene of some of the war's bloodiest battles, and force them to surrender or withdraw.
"Despite significant losses, the enemy threw in the most prepared assault units of Wagner, who are trying to break through the defenses of our troops and surround the city," Ukraine's Colonel General Oleksandr Syrskyi said.
Ukraine's military said Russia was shelling settlements around Bakhmut, which had a pre-war population of around 70,000 but now lies in ruins after months of intense trench warfare.
"Over the past day, our soldiers repelled more than 60 enemy attacks," the Ukraine military said early on Tuesday, including on the villages of Yadhidne and Berkhivka just north of Bakhmut.
A Reuters reporter who visited the area on Monday said he saw no sign of Ukrainian forces withdrawing and that reinforcements were arriving despite constant Russian shelling.
Taking Bakhmut would give Russia its first major prize in more than half a year and open the way to the capture of the last remaining urban centers in the Donetsk region, one of four which Moscow claims to have annexed in what it calls its "special military operation" in Ukraine.
An unnamed soldier from Ukraine's 93rd Separate Mechanized Brigade, speaking on the Telegram messaging app as explosions boomed in the background, struck a defiant note: "February 28, the town of Bakhmut. The city is on fire, the enemy is pressing. Everything will be Ukraine..."
Russia's state-run RIA news agency released a video clip which it said showed Russian Su-25 fighter jets roaring over Bakhmut. "We are glad they are ours," says a man in the clip identified as a Wagner fighter, adding the jets helped them "psychologically."
Ukrainian soldiers in the Donetsk region hunkered in muddy trenches after warmer weather thawed out the frozen ground.
"Both sides stay in their positions, because as you see, spring means mud. Thus, it is impossible to move forward," said Mykola, 59, commander of a Ukrainian frontline rocket launcher battery, watching a tablet screen for coordinates to fire.
The spring thaw has a history of ruining plans by armies to attack across Ukraine and western Russia, turning roads into rivers and fields into quagmires.
Reuters saw several military vehicles stuck in mud. In a zigzag trench, Volodymyr, a 25-year-old platoon commander, said his men were prepared to operate in any weather.
"When we're given a target, that means we have to destroy it."
Russia, its forces replenished with hundreds of thousands of conscripts, has intensified its attacks right along the eastern front but its assaults have come at a high cost, Ukraine says.
Russia's defense ministry said its forces had destroyed a Ukrainian ammunition depot near Bakhmut and shot down U.S.-made rockets and Ukrainian drones.