Citing sources in Ukraine's military intelligence (GUR), the agency said the hackers had bypassed "all Russian protection systems" and that attacks were expected until the end of the vote.
The head of Russia's Central Election Commission (CEC), Ela Pamfilova, said that at least 10,500 attacks against the voting system have been blocked since Friday.
The United Russia party has also reported that its electronic services are suffering large-scale attacks.
The election is expected to keep Putin in power until 2030, the year he turns 77, with the possibility of an additional term until 2036 due to a constitutional change made in 2020.
The war triggered by the Russian invasion of Ukraine plunged Europe into what is considered the worst security crisis since the Second World War (1939-1945).
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