Russia pressed its attacks in Ukraine's eastern Donbas region, where Ukrainian President Volodymyr Zelenskiy said the situation remained "extremely difficult".
Russia has been seeking to seize the entire Donbas, consisting of Luhansk and Donetsk which Moscow claims on behalf of separatist proxies.
Capturing the twin cities of Sievierodonetsk and Lysychansk on the banks of the Siverskyi Donets river would give Moscow effective control of Luhansk and allow the Kremlin to declare some form of victory after more than three months of the war.
But by focusing on a battle for the single small city, Russia could leave other territories open to Ukrainian counterstrikes.
Kyiv said its forces had pushed back Russian troops to defensive positions in Andriyivka, Lozove and Bilohorka, villages on the south bank of the Inhulets River that forms the border of Kherson province, where Moscow is trying to consolidate control.
Ukraine has called for the West to send more long-range weapons but US President Joe Biden said Washington would not send Ukraine rocket systems that can reach into Russia, a decision Russian Security Council Deputy Chairman Dmitry Medvedev called "rational."
Zelenskiy said Russian forces shelled the northeastern city of Kharkiv again on Monday, as well as the border region of Sumy, which was hit from inside Russia.
Russian shelling has reduced much of Sievierodonetsk to ruins, but the Ukrainian defence has slowed the wider Russian campaign across the Donbas region.
Luhansk governor Serhiy Gaidai said Russian troops had advanced into Sievierodonetsk's southeastern and northeastern fringes, but Ukrainian forces had driven them from the village of Toshkivka to the south, which could frustrate a push to encircle the area.
"They use the same tactics over and over again. They shell for several hours - for three, four, five hours - in a row and then attack," he said. "Those who attack die. Then shelling and attack follow again, and so on until they break through somewhere."
With temperatures rising, there was a "terrible smell of death" on the outskirts of Sievierodonetsk, Gaidai said.
The leader of the Moscow-backed Luhansk People's Republic, Leonid Pasechnik, told Tass news agency a third of Sievierodonetsk was "already under our control".
A French journalist, Frederic Leclerc-Imhoff of television channel BFM, was killed near Sievierodonetsk on Monday when shelling hit the vehicle he was travelling in during an evacuation of civilians. French Foreign Minister Catherine Colonna, who was visiting Ukraine, demanded an investigation.
Russian President Vladimir Putin, in talks with Turkish President Tayyip Erdogan, said Russia was ready to facilitate unhindered grain exports from Ukrainian ports in coordination with Turkey, according to the Kremlin.
Western leaders have chided Russia for blockading Ukrainian ports, sending prices of grain and other commodities soaring. The United Nations has said a global food crisis is deepening and has been trying to broker a deal to unblock Ukraine's grain exports.
"Emphasis was placed on ensuring safe navigation in the Black and Azov seas and eliminating the mine threat in their waters," the Kremlin said of Putin's call with Erdogan.
Putin said if sanctions were lifted, Russia could export significant volumes of fertilisers and agricultural products.
Zelenskiy also spoke with Erdogan and said they discussed food security and defence cooperation, "and, of course, how to accelerate the end of this war."