According to the ministry, Ukrainian "sabotage and reconnaissance" teams were stopped while trying to cross the river near the villages of Pridniprovske, Tiahynka and Krynky.
Russia also said it had destroyed Ukrainian personnel, water-crossing equipment and vehicles near the village of Stanislav.
Reuters could not independently verify battlefield reports.
On Friday the Institute for the Study of War, a US research group, said Ukrainian forces appeared to have broken through on the eastern bank of the Dnipro River in Kherson.
Ukraine recaptured parts of the Kherson region late last year after months of Russian occupation.
But Russian forces who left Kherson, the region's biggest city, retreated only as far as the other side of the Dnipro and have continued to shell the city from there.
Three Ukrainian missiles heading for targets in Crimea were downed on Sunday over the Kherson region, local governor Vladimir Saldo said.
Saldo, writing on the Telegram messaging app, said the missiles were downed over the town of Henichesk.
Russia annexed Crimea in 2014.
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