The move, while not unexpected, is one of Russia's most pronounced nuclear signals yet and a warning to NATO over its military support for Ukraine, which has called for a meeting of the United Nations Security Council in response.
"Russia's nuclear rhetoric is dangerous and irresponsible," NATO spokeswoman Oana Lungescu said on Sunday.
"NATO is vigilant and we are closely monitoring the situation.
"We have not seen any changes in Russia's nuclear posture that would lead us to adjust our own."
Putin likened his Belarus plan on Saturday to the United States stationing its weapons in Europe, insisting Russia would not violate its nuclear non-proliferation promises.
However, Lungescu said Putin's non-proliferation pledge and his description of US weapons deployment overseas were way off the mark.
"Russia's reference to NATO's nuclear sharing is totally misleading. NATO allies act with full respect of their international commitments," she said in a statement.
"Russia has consistently broken its arms control commitments."
Ukraine security chief Oleksiy Danilov said Russia's plan would destabilise Belarus, which he said had been taken "hostage" by Moscow.
Others condemning Putin's plan included Lithuania, which said it would call for new sanctions against Moscow and Minsk, while EU policy chief Josep Borrell urged Belarus not to host the weapons and threatened more sanctions.
Belarus and Russia have close military ties, and Minsk allowed Moscow to use its territory as a staging point for the latter's invasion of Ukraine last year.
Experts see Russia's move as significant since it had been proud, until now, of not having deployed nuclear weapons outside its borders, unlike the US. This may be the first time since the mid-1990s that it plans to do so.
The US, also a nuclear superpower, played down concerns about Russia's planned deployment.
"I can tell you we've seen nothing that would indicate Mr Putin is preparing to use tactical nuclear weapons in any way whatsoever in Ukraine," White House National Security Council spokesman John Kirby told broadcaster CBC on Sunday.
"And I can also tell you that we haven't seen anything that would cause us to change our own strategic nuclear deterrent posture."
On the battlefield, Russian shelling forced an almost full shutdown of the eastern Ukrainian city of Avdiivka, just 90km southwest of the besieged Bakhmut, the top regional official said on Sunday.
"I am sad to say this but Avdiivka is becoming more and more like a place from post-apocalyptic movies," said Vitaliy Barabash, head of the city's military administration head.
Russian shelling targeted two high-rise buildings in Avdiivka, 10km from the northern outskirts of the city of Donetsk that has been under Moscow's control since 2014.
Last week, the Ukrainian military warned Avdiivka could become a "second Bakhmut" - which has been reduced to rubble in fierce fighting that both sides have called a "meat grinder".
Russia is pressing on with its assault on Bakhmut, while Ukrainian forces repelled more than 60 attacks in the past 24 hours across the eastern front, some in the direction of Avdiivka, Ukraine's general staff said on Monday.
Russia's defence ministry said it had downed a Ukrainian drone south of Moscow on Sunday, adding three people were injured and apartment blocks were damaged in the drone attack.
Russia has said in the past that Ukrainian drones have flown into its territory and damaged civilian infrastructure, an assertion Kyiv denies.
Reuters could not immediately verify the battlefield reports.
