Calling any attack on a nuclear plant "suicidal", United Nations chief Antonio Guterres demanded UN nuclear inspectors be given access to Zaporizhzhia, the largest complex of its kind in Europe.
Russia's invading forces seized the southern Ukrainian region containing Zaporizhzhia in March, when the site was struck without damage to its reactors.
The area, including the city of Kherson, is now the target of a Ukrainian counteroffensive.
Ukraine appealed for the area around the complex to be demilitarised and for the International Atomic Energy Agency, the UN's nuclear watchdog, to be let in.
Russia said it also favoured an IAEA visit, which it accused Ukraine of blocking while trying to "take Europe hostage" by shelling the plant.
Ukraine blamed Russia for weekend attacks around the complex, which is still being run by Ukrainian technicians. It said three radiation sensors were damaged and two workers injured by shrapnel.
As of Monday morning, the plant appeared to still be running, said Petro Kotin, head of Ukraine's state nuclear power company Energoatom.
He said 500 Russian soldiers and 50 pieces of heavy machinery, including tanks, trucks and armoured infantry vehicles, were at the site.
Kotin called for peacekeepers to run the Zaporizhzhia site, flagging the risk of shells hitting its six containers of highly radioactive spent nuclear fuel.
In an evening video shared online, Ukrainian President Volodymyr Zelenskiy called for new Western sanctions on Russia's nuclear industry "for creating the threat of a nuclear disaster".
Dr Mark Wenman, a nuclear expert at Imperial College London, played down the risk of a major incident, saying the Zaporizhzhia reactors were relatively robust and the spent fuel well protected.
"Although it may seem worrying, and any fighting on a nuclear site would be illegal ... the likelihood of a serious nuclear release is still small," he said in a statement.
Yevhenii Tsymbaliuk, Ukraine's ambassador to the IAEA, said Zaporizhzhia staff were "working under the barrels of Russian guns".
Meanwhile, Russia's defence ministry said Ukrainian attacks had damaged power lines servicing the plant and forced it to reduce output by two of its six reactors to "prevent disruption".
The UN's Guterres said IAEA personnel needed access to "create conditions for stabilisation".
"Any attack (on) a nuclear plant is a suicidal thing," he told a news conference in Japan, where he attended the Hiroshima Peace Memorial Ceremony on Saturday to commemorate the 77th anniversary of the world's first atomic bombing.
Ukraine has said it is planning to conduct a major counteroffensive around Kherson and that it has already retaken dozens of villages.
Its forces are also fighting to retake areas near Kharkiv in the north, where Russian forces launched artillery strikes on Monday, Ukraine's general staff said.
In Ukraine's Donetsk region, where pro-Moscow separatists seized territory after the Kremlin annexed Crimea to the south in 2014, Russia was "using all available fire power ... to try and inflict maximum losses on Ukrainian units to prevent them from reinforcing other areas", the general staff said.
In Washington, the Pentagon stepped up its military spending on Ukraine, pledging $US1 billion ($A1.4 billion) of further security assistance, including munitions for long-range weapons.
Russia's foreign ministry, meanwhile, told the United States it was suspending inspection activities under their START nuclear arms control treaty, though it said Moscow remained committed to the treaty's provisions.
Elsewhere, a deal to unblock Ukraine's food exports and ease global shortages gathered pace as two grain ships carrying almost 59,000 tonnes of corn and soybeans sailed out of Ukrainian Black Sea ports.
That raised the total to 12 since the first vessel left a week ago.
Russia says it is waging a "special military operation" in Ukraine to rid it of nationalists and protect Russian-speaking communities. Ukraine and the West describe Russia's actions as an unprovoked war of aggression.
