While many of Kyiv’s battlefield claims could not be verified, it is clear that Ukrainian forces have made minor gains in recent weeks to the north and south of Bakhmut, breaking through a Russian defensive position in a key battle that started on May 6.

Speaking during a visit to Ukrainian soldiers on the eastern front on Sunday, General Syrsky claimed on Sunday that Ukrainian forces were continuing to make advances in the suburbs of Bakhmut and were “close to tactically encircling the city.” But the Ukrainian military did not provide details about where it was advancing.

On Monday, Maj. Rodion Kudriashov, a commander in the Ukrainian 3rd Assault Brigade, said the Ukrainians continued to make small but important gains around the city.

“We are trying to feel the enemy’s weak spots and gradually, step by step, we are trying to squeeze him out, occupying advantageous positions for us, taking the initiative,” he said in an interview with the Ukrainian broadcaster RBC-Ukraine.

Some analysts said the Ukrainian claims appeared to be an effort to put a gloss on the setback in Bakhmut.

“Talks about tactical encirclement are premature and probably reflect a hasty response to the city’s fall,” Konrad Muzyka, a defense analyst for Rochan Consulting, wrote on Monday.

That does not mean, however, that Russia’s grip on the city is assured. A great deal could depend on the plans of the mercurial and bombastic head of Russia’s Wagner mercenary group, Yevgeny Prigozhin, whose fighters spearheaded the Russian assault on Bakhmut. Hours after declaring “victory” over the weekend, Mr. Prigozhin said he would withdraw his fighters beginning on Friday.

“From June 1, not a single Wagner PMC fighter will be at the forefront until we undergo re-formation, re-equipment and additional training,” he said.

Withdrawing forces from an active front is no simple task, and could leave Russia vulnerable to attacks from Ukrainian troops on high ground on the city’s outskirts. Given the widely reported tensions between Wagner and Russia’s military leadership, and communication problems within the Russian ranks, analysts say Ukraine will be watching for fissures to exploit.

Even if Mr. Prigozhin does not pull his fighters out now, military analysts noted that Russia will face challenges controlling Bakhmut, especially with Ukrainian forces preparing a counteroffensive that could escalate fighting along other parts of the 600-mile-long front line.

“Russian forces will likely need additional reinforcements to hold Bakhmut city and its flanks at the expense of operations in other directions,” the Institute for the Study of War wrote in its latest analysis on Sunday.

The Ukrainians and British military intelligence have both reported that there is evidence the Russian military is sending thousands of soldiers to the area.

Hanna Maliar, a deputy Ukrainian minister of defense, said on Monday that while “the enemy is mopping up the areas of the city under his control,” Ukraine is continuing “the struggle for dominant heights” around the city.

For now, she said, Ukrainian soldiers had fulfilled their primary objective in the battle for Bakhmut.

“The offensive potential of the enemy is significantly reduced,” she said. “The enemy has suffered huge losses, and we have gained time for certain actions that can be described later.”