Kyiv city officials said air defences were at work and the air force reported Russia had launched hypersonic missiles at the Kyiv region.
"Explosions in the city. Stay in shelters!" Klitschko wrote on the Telegram messaging app.
He said fragments of a missile had come down on the territory of a children's hospital in the city, but there were no injuries or damage.
Reuters journalists in the capital could hear explosions but could not immediately verify the precise location.
National television broadcaster Suspilne reported explosions in the central and western regions of Vinnytsia and Khmelnytskyi.
An official in Khmelnytskyi said air defences in the area were at work.
In Russia, the airspace over Moscow's Vnukovo airport was temporarily closed on Friday, with arrivals and departures suspended.
Russia's RIA news agency said the move was due to a suspected drone flight in the area.
"For reasons beyond the control of the airport, temporary restrictions on the landing and take-offs of aircraft were introduced in Vnukovo," the airport said.
"For safety reasons, some of the flights were redirected to other airports of the Moscow aviation hub," it said, without commenting further.
There have been a number of drone attacks in the Moscow area in recent weeks that Russia has blamed on Ukraine.
The attacks come as the German government holds talks with arms maker MBDA about the delivery of Taurus cruise missiles to Ukraine, a security source told Reuters on Friday, confirming a report by Spiegel magazine.
Kyiv has been pushing Berlin to supply it with the Taurus, a missile with a range of more than 500 kilometres that is fired from fighter jets such as the Tornado, the F-15 or the F-18.
On Thursday, President Joe Biden asked Congress to approve about $US40 billion ($A61 billion) in additional spending including $US24 billion ($A37 billion) for Ukraine.
The request could face opposition in Congress, where some far-right Republicans - especially those with close ties to former president Donald Trump - want to pare back the billions in assistance Washington has sent to Kyiv since Russian troops invaded in February 2022.
Trump has been sharply critical of US support for Ukraine in the war and 70 House members backed an unsuccessful proposal in July to cut funding for Ukraine.
Senate Majority Leader Chuck Schumer said there was strong bipartisan support in the Democratic-led Senate for doing more to back Ukraine.
Republicans narrowly control the House, where Speaker Kevin McCarthy signalled in June that any request for more assistance for Ukraine would face an uphill path through Congress.
A spokesperson for McCarthy said he would review Biden's proposal.
"A Republican-led House will not rubber-stamp any blank-check funding requests; rather, the administration's emergency funding requests must be reviewed and scrutinised on their merits consistent with the practice and principles of our majority," the spokesperson said in an emailed statement.
The House and Senate last approved aid for the Kyiv government - $US48 billion ($A74 billion) - in December, before Republicans took control of the House.
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