In its morning update the president's office in Kyiv said "heavy fighting is taking place in almost the entire territory of the Kherson region".
"Powerful explosions continued throughout the day and throughout the night" and "the Armed Forces of Ukraine launched offensive actions in various directions," it added.
In his nightly address on Monday, President Volodymyr Zelensky was coy about Ukraine's overarching strategy whilst striking a bullish tone.
"You won't hear the specifics from any person in charge because this is war."
"But the occupiers should know that we will push them to our borderline," he added.
"If they want to survive, it is time for the Russian military to flee. Go home."
Moscow's troops seized Kherson, a town of 280,000 inhabitants, on March 3. It was the first major city to fall following Moscow's invasion of Ukraine on February 24.
In an update early Tuesday, Ukraine's Southern Command said the situation remained "tense" in its area of operations.
"The enemy attacked our positions five times, but was unsuccessful," it said.
The city of Mykolaiv, just northwest of Kherson, had come under "massive bombardment" from Russian anti-aircraft missiles, with two civilians killed and 24 wounded, it said.
On Monday, Russia's defense ministry claimed it had repulsed attacks in the Kherson and Mykolaiv regions and inflicted "heavy losses" on Ukrainian forces.
The ministry said Ukraine's troops attempted attacks in "three directions" but that they lost more than 560 servicemen and 26 tanks.
But a senior Pentagon official said Russia was struggling to find soldiers to fight in Ukraine and that many new recruits were older, in poor shape and lacking training.
Britain's defense ministry said starting early Monday "several brigades of the Ukrainian Armed Forces increased the weight of artillery fires in front line sectors across southern Ukraine".
It warned that since the start of August, Russia has made "significant efforts" to reinforce troops on the western bank of the Dnipro River which splits Kherson city.
But the Ukrainian presidency claimed its forces had destroyed "almost all large bridges" and that "only pedestrian crossings remain" in Kherson region.
The British ministry said "most of the units around Kherson are likely under-manned and are reliant upon fragile supply lines by ferry and pontoon bridges across the Dnipro".
"It is not yet possible to confirm the extent of Ukrainian advances," the ministry added on Tuesday.
Most of region of Kherson bordering the Black Sea -- and its provincial capital of the same name -- were seized by Moscow's military at the start of the invasion six months ago.
With the war in the eastern Donbas region largely stalled, analysts have said for weeks that combat is likely to shift south to break the stalemate before winter comes.