As the Russian attacks hit Kyiv through the early morning, the buzz of drones could be heard between more than a dozen loud explosions as strikes made impact.
The attack caused fires near a gas station, a construction site, and several apartment blocks, as well as two houses, Klitschko said.
Blackouts and fires have also been reported across the city and the wider extent of damage is not yet clear.
An industrial facility has also been attacked in Zaporizhzhia.
The attacks come after Ukrainian President Volodymyr Zelensky on Monday reiterated warnings of a possible large Russian strike and urged residents to pay special attention to air raid alerts.
“Intelligence warnings regarding Russian strikes remain in effect. A massive strike is possible, they have prepared one,” Zelensky said in his nightly video address.
Last week Russia said it would target military and decision-making centres in Kyiv, and urged foreigners to leave the city in what it said was a response to Ukraine deliberately striking a dormitory in Strabolinsk.
In a statement at the time, the General Staff of Ukraine’s Armed Forces said it did carry out an attack near Starobilsk on the night of 21-22 May, but maintains that it struck a Russian military unit.
Kyiv said Russia’s threats were “nothing short of shameless blackmail” and urged allies to increase pressure on Moscow.
Since a brief ceasefire expired in May, Russia has launched several waves of missile and drone attacks on Kyiv, among them strikes on a block of flats that killed 24 people, including three children.
A Ukrainian drone attack on the Moscow region killed three people, in what Zelensky said was an “entirely justified” response to Russian attacks.