Ukraine's military said fierce fighting was underway in a part of the city that was key for its frontline logistics.
It said additional special forces had arrived there, and more weapons and equipment were being sent.
The Russian Defence Ministry said its soldiers had cleared 35 buildings of Ukrainian troops.
It also said that Russian forces were squeezing surrounded Ukrainians near the town of Kupiansk in the Kharkiv region, about 160km to the north.
Reuters was unable to verify the battlefield reports from either side.
Ukraine has denied that its troops are encircled in either location.
With peace talks at an impasse in the fourth year of the war, Russia says it has the initiative on all sections of the front yet it has not captured a major town since taking Avdiivka in early 2024.
Ukraine says the fighting is largely stalemated and its territorial losses have been marginal.
DeepState, a Ukrainian project that maps the front line based on verified open source images, on Tuesday showed that Russian forces had pushed further into Pokrovsk and its environs although it showed much of it still in grey - beyond firm control of either side.
Pokrovsk had a pre-war population of about 60,000 but most civilians fled long ago.
Capturing it could give Russia a platform to drive towards Kramatorsk and Sloviansk, the two biggest remaining Ukrainian-controlled cities in the Donetsk region which Russia wants to capture in its entirety.
Russian military blogger Rybar said on Tuesday that the Kremlin's control of Pokrovsk was gradually expanding but "a complete clearing of the city is still far off".
Ukrainian President Volodymyr Zelenskiy acknowledged on Monday that Pokrovsk was under severe pressure but his military said Russian troops were not in full control of any district.
Zelenskiy said on Tuesday he had visited troops fighting near the eastern city of Dobropillia, where Ukrainian forces are conducting a counteroffensive against Russian troops.
"Much attention was given to weapons, scaling up drone production, the needs of brigades and the experience of the 1st Corps of the National Guard of Ukraine 'Azov' and other units," he wrote on X.
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