Ukrainian President Volodymyr Zelensky's chief of staff Andriy Yermak has said that Ukrainian forces have gained a foothold on the left or east bank of the Dnipro river.
Russia has admitted that some Ukrainian forces have set up positions in a village but claims they will soon be wiped out.
If the area is held it would mean a significant advance for Ukraine.
Russia retreated from the right bank of the river a year ago.
The Ukrainian army has since been trying to create and hold a bridgehead on its eastern shore.
"The Armed Forces of Ukraine captured a bridgehead on the left bank of the Dnipro," Mr Yermak told a US think tank, using a military term for a secured position.
US-based experts said earlier that marginal advances had been made into the village of Krynky, 2km (1.25 miles) inland from the river and 30km from the city of Kherson, recaptured a year ago.
Russia conceded on Wednesday that "small groups" of Ukrainian forces had set up positions in the village on the left bank.
But Vladimir Saldo, the Russian-installed official in charge of the Kherson region still under occupation, insisted that they had sustained heavy losses and faced a "fiery hell" and had no chance of breaking through. "Our additional forces have now been brought in. The enemy is trapped in Krynky," he claimed.
His admission that Ukrainian forces had gained a foothold came two days after Russian state media bizarrely published and then swiftly retracted news of a retreat from positions on the left bank.
Repeated attempts to cross the river made during the spring and summer had been met with little success.
Ukrainian forces conducted several raids across the Dnipro in small boats. However, Kyiv lacks air superiority, which made the task more difficult.
Securing a bridgehead means that Ukrainian forces may be able to begin transferring armoured vehicles and air defence systems across the river. This would put them one step closer to breaking through to Crimea, the peninsula illegally annexed by Russia in 2014.
"Step by step, Ukraine's armed forces are demilitarising Crimea. We have covered 70% of the distance," Mr Yermak added. He appealed to Western countries to provide more weapons to Kyiv, saying Russia was using Iranian drones and North Korean artillery against Ukrainian forces.
On Tuesday, President Zelensky said fighting was escalating in the east near the town of Avdiivka. He added, however, that Russian forces were "losing men and equipment" trying to capture the destroyed city.
Ukraine launched its much-anticipated counteroffensive in June, seeking to regain territory in the south and east. However, the counteroffensive has so far failed to recapture significant territory.
