President Zelenskyy has said that Ukrainian forces are slowly moving towards the Russian-held city of Kherson.
Ukrainian troops have damaged bridges in Kherson that act as supply lines for Russian troops.
UK military intelligence has said that losing supply routes to Kherson "would be a significant military and political setback for Russia."
President Volodymyr Zelenskyy has said that Ukrainian forces are slowly moving towards the Russian-held city of Kherson, according to the BBC.
The President's announcement comes as Ukrainian forces claim they have damaged two important bridges across the Dnipro River vital to the lines of supply to Russian troops in and around Kherson, Reuters reports.
The British Ministry of Defense said that "Supply lines of the Russian forces west of the river are increasingly at risk" and that if the Russian forces in Kherson do become cut off, "it would be a significant military and political setback for Russia."
On Saturday, UK defense officials reported heavy fighting near Kherson.
The Antonovsky and the Daryivskyi bridges across the Dnipro River, connecting Kherson to southern Ukraine and Crimea, have been damaged by Ukrainian shelling, according to reports.
The deputy head of the Russian-installed regional authority said the Daryivskyi bridge had been hit by seven rockets from US-supplied high mobility artillery rocket systems (HIMARS) on Saturday, but that the bridge still worked, Russia's TASS news agency said, per Reuters.
Serhiy Khlan, an advisor to the region's governor, wrote on Facebook: "Every bridge is a weak point for logistics and our armed forces are skilfully destroying the enemy system. This is not yet the liberation of Kherson, but a serious preparatory step in that direction."
Ukrainian state media channel Euromaidan quoted presidential advisor Oleksiy Arestovych as saying there would be no humanitarian (or "green") corridor for Russian troops to leave Kherson.
"We have a very strong temptation to bury them all," he said before adding, "but, first of all, we do not fight like the Russians, we do not break our word. -secondly, I think that the highly humane Ukrainian Armed Forces will simply offer them to surrender."
"We will not provide a corridor. If they do not want to surrender, their fate will be decided," he said.
Dr. Mike Martin, War Studies Visiting Fellow at King's College London, tweeted that "if I were a Russian soldier in Kherson, I would be pretty scared right now."
He added, "I would be watching Kherson very closely over the next ten days."
Russian forces seized Kherson, a strategically important port city of 300,000 people, on 3 March. It was the first major city to fall following Moscow's invasion of Ukraine.