Kyiv said its troops were fighting to retain control of the now-battered industrial towns in the east, which Russian mercenaries claimed to have taken this week.
The United Nations Security Council was due to meet at 2000 GMT on Friday to discuss the situation in Ukraine.
The Kremlin has made capturing Bakhmut — and Soledar with it — its primary objective after nearly one year of fighting, having been forced to abandon more ambitious goals such as seizing the capital Kyiv.
“I want to emphasise that the units defending these cities will be provided with ammunition and everything necessary, on time and without interruption,” Zelenskyy said in a statement on Thursday after a meeting with senior military officials.
Ukraine’s Deputy Defence Minister Ganna Malyar said earlier that the fight for Soledar was “the fiercest and heaviest” of the war.
“Despite the difficult situation, Ukrainian soldiers are fighting stubbornly,” she added.
Although Russian mercenary group Wagner claimed early Wednesday that its forces had captured Soledar, the defence ministry in Moscow said fighting was ongoing and Ukraine denied any full takeover.
Military maps released by Russia‘s defence ministry on Thursday did not show Soledar under the control of Moscow’s regular army.
A Russia-installed official in Donetsk, Andrey Baevsky, said there were still “small pockets of resistance” from Ukraine inside the city, claiming Russian-backed troops had nearly full control.
Both sides have conceded heavy losses in the fight for Soledar and the nearby larger town of Bakhmut, which is also key to Russia’s aim to wrest all of Donetsk away from Ukraine.
The Kremlin on Thursday praised the “heroic” work by Russian forces working to capture the eastern Donetsk region from Ukraine and on other fronts.
“Huge work has been done in Soledar, absolutely selfless heroic actions, not only in Soledar,” Kremlin spokesman Dmitry Peskov said.
“There is still a lot of work ahead. The main work is yet to come,” he added.
The battle for Soledar comes as Moscow on Wednesday announced a major military reshuffle, putting Chief of the General Staff Valery Gerasimov in charge of its operations in Ukraine.
A Moscow-based defence analyst, who spoke on condition of anonymity, described the move to AFP as “unprecedented” and said it indicated “very serious problems” on the battlefield.
“This has not happened since 1941, when Marshal Georgy Zhukov was sent to the front to command.”
As part of Wednesday’s appointments, the head of Russia’s ground forces, Oleg Salyukov, was named a deputy commander of Russia’s military campaign in Ukraine.
On Thursday, he visited Moscow ally Belarus to inspect a joint regional force stationed there, Russia’s defence ministry said.
When Russia invaded in February, Belarus allowed Moscow’s troops stationed there to cross the Ukraine border via its territory.
Russia wants to gain control of the Donetsk region, which it claimed to have annexed last year despite not having complete control over it.
The conflict in Ukraine has come to be defined by the use of drones ranging from small commercially-available models to larger aircraft, with both sides trying to outmanoeuvre each other.
“Both Russians and Ukrainians are now saying publicly that there are parts of the front where their military drones cannot operate, where their commercial drones can be jammed and rendered inoperable,” Pentagon press secretary Brigadier General Pat Ryder said.
Observers have said that Soledar itself — a salt mining town with an estimated pre-war population of more than 10,000 people — is of little strategic importance.
Its capture, however, would allow Russia to sell a much-needed victory back home after months of humiliating battlefield reversals.
“Any victory is important, especially because there hasn’t been a victory in a while,” Russian military analyst Anatoly Khramchikhin told AFP.
In Bakhmut, which has been shelled daily for months by Russian forces trying to capture it, one of the few remaining doctors still working there told AFP she was determined to stay.
“When I enrolled in medical school, I took the Hippocratic oath, and I cannot betray these people,” said Elena Molchanova.
“As long as they are here, I’ll be here.”
Her work mainly consisted of distributing meagre medical supplies to some 8,000 residents still in the war-scarred city and handing out death certificates, she added.