Kherson city fell to Russia early in the war and sits on a strategic location west of the Dnipro river.
On Saturday, UK defence officials reported heavy fighting near Kherson.
The Ukrainian advance meant that Russian supply lines west of the river were “increasingly at risk”, according to their assessment.
Earlier this month, Ukraine’s Deputy Prime Minster Iryna Vereshchuk urged Kherson residents to evacuate the city as soon as possible to avoid becoming trapped in the city during a counteroffensive in southern Ukraine.
“It is necessary to do so for the Armed Forces of Ukraine not to endanger the civilian population during offensive operations,” she told state TV.
Serhiy Khlan, an adviser to Kherson’s government, told Ukrainian television the region would “definitely be liberated by September”, AFP report.
Kyiv’s forces have been targeting river crossings in the region in an effort to stretch Russia’s supply lines. On Saturday, an artillery strike hit the Daryivskyi bridge across the Inhulets river, a tributary of the Dnipro.
And on Tuesday, they targeted the Antonivskyi Bridge over the Dnipro itself with US-supplied artillery.
Speaking to the state backed Tass news agency, the deputy head of the Russian-backed administration in Kherson admitted that if the strikes continued the bridge could collapse.
UK officials have described the Antonivskyi Bridge as a “key vulnerability” for Russian forces and on Saturday said if “crossings were denied, and Russian forces in occupied Kherson cut off, it would be a significant military and political setback for Russia”.
Mr Khlan said “every bridge is a weak point for logistics and our armed forces are skilfully destroying the enemy system”.
Meanwhile, a senior defence adviser to Mr Zelensky claimed that around 1,000 Russian troops in the region have been encircled by Ukrainian forces.
Oleksiy Arestovych said the Russians had been caught in a “tactical encirclement” near the village of Vysokopillya in Kherson oblast. The BBC has not independently verified this claim.
Tass reported that plans in Kherson to hold a referendum on joining Russia have moved forward, with authorities there forming an election committee. The US has accused Russia of preparing to annex parts of Ukraine.
Russia captured Kherson with relatively little resistance early in the war, and the failure of Ukraine’s security service (SBU) to destroy crossing points over the Dnipro before fleeing the city is believed to have led to Mr Zelensky’s dismissal of the agency’s director Ivan Bakanov on Monday.
