Russia is observing a day of mourning after at least 133 people were killed on Friday evening in an attack at a packed concert venue in Moscow.
More than 140 people were also injured when gunmen entered the Crocus City Hall, firing indiscriminately before setting it on fire.
The Islamic State (IS) group says it was behind the attack.
In a televised address on Saturday, Mr Putin condemned the massacre – the deadliest in Russia for nearly 20 years – as a "barbaric terrorist act" and repeated earlier suggestions by Russian security services that the attackers had tried to escape to Ukraine.
Kyiv dismissed the claim that it was in some way involved in the attack as "absurd".
It comes as strikes on Ukraine continue over the weekend with a million people without power due to the targeting energy infrastructure.
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