Ukraine is reinforcing the area with elite units, while regular Russian army and forces of the Russian private military Wagner group have made further advances into Bakhmut's northern suburbs, the British Defence Ministry said in its daily intelligence bulletin on Twitter.
Two key bridges in Bakhmut have been destroyed within the last 36 hours, it said on Saturday, adding that Ukrainian-held resupply routes out of the city are increasingly limited.
Ukraine's military command said Russia was still trying to surround Bakhmut but added that over the past day Ukrainian forces had beaten back Russian attacks in the city.
"The enemy does not cease attempts to surround Bakhmut," it said in its morning briefing note on Saturday.
Russian artillery pounded the last routes out of Bakhmut on Friday, aiming to complete the encirclement of the besieged city and bring Moscow closer to its first major victory in the war in six months.
The Ukrainian briefing note also said Russian attacks had been foiled in the villages of Ivanivske and Bohdanivka, both of which lie less than 8km west of Bakhmut's city centre.
The capture of those villages, which flank the crucial Bakhmut-Chasiv Yar road on either side, would leave the city on the cusp of total Russian encirclement.
The battle for Bakhmut has raged for seven months. A Russian victory in the city, which had a pre-war population of about 70,000 and has been blasted to ruins in the onslaught, would give Moscow the first major prize in a costly winter offensive, after it called up hundreds of thousands of reservists last year.
Russia says it would be a stepping stone to completing the capture of the Donbas industrial region, one of Moscow's most important objectives.
Ukraine's President Volodymyr Zelenskiy has described Bakhmut as a "fortress".
"Nobody will give away Bakhmut. We will fight for as long as we can. We consider Bakhmut our fortress," he told a news conference in Kyiv on February 3.
