Ukraine’s president Volodymyr Zelenskyy had responded: “We don’t attack Putin or Moscow – we fight on our own territory”.
The Kremlin said the drones were disabled before they could strike and there were no reports of victims or damage.
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Ukraine’s military said the three Russian-launched drones that hit a university in Odesa in the early hours of Thursday had been inscribed “for Moscow” and “for the Kremlin”.
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Zelenskyy leads silence at The Hague
The Kremlin had said it reserved the right to retaliate for the attack, but it had not said what form this retaliation might take.
Mr Peskov said on Thursday that Russia has several options and that the response – when it comes – will be carefully considered and balanced.
His words came just hours after Ukraine said it had shot down more than a dozen Russian-launched drones in the early hours of Thursday morning.
The 18 drones were among 24 launched in pre-dawn attacks across Ukraine, with a number aimed at Kyiv – all of which were destroyed alongside an unspecified number of missiles, city officials said.
Administration officials in the capital said: “The Russians have attacked Kyiv using Shahed loitering munitions and missiles, likely the ballistic type.”
Some 15 Shahed drones were fired at the Black Sea port city of Odesa, with air defences destroying 12 of these, according to Ukraine’s southern military command.
Three of the drones struck a university compound. There were no casualties.
Ukrainian public broadcaster Suspline also reported explosions in the southern city of Zaporizhzhia, with the head of the city’s regional military administration Yuri Malashko confirming anti-aircraft defences were at work.
Russia has regularly bombarded Ukraine since October and the latest attacks come less than 24 hours after Kyiv said 21 people were killed in a Russian strike on the city of Kherson.
Mr Zelenskyy said a supermarket, railway station and residential buildings were among the places hit.
The Ukrainian president visited the International Criminal Court at The Hague on Thursday morning and reiterated his denials of involvement in the attack on the Kremlin.
He said: “We didn’t attack Putin. We leave it to (the) tribunal.”
Speaking at a news conference in the Netherlands, Mr Zelenskyy said he hoped the Russian leader would face an international war crimes court in the future.
“We all want to see… Vladimir here in The Hague,” he said, adding he “deserves to be sentenced” and brought to justice.