The Ukrainian source said Kirillov - who was Russia's chemical weapons chief - was "a legitimate target" and alleged he had carried out war crimes.
On Monday, the day before the killing, Ukraine charged Kirillov, 54, in absentia, saying he was "responsible for the mass use of banned chemical weapons". Russia denies the allegations.
A Kremlin spokesman said Russian President Vladimir Putin "expresses deep condolences" over Kirillov's death, Russian news agency Tass reported.
The Russian Federal Security Service's (FSB) public relations centre said on Wednesday the detained 29-year-old was "suspected of committing a terrorist act".
A statement said that during "interrogation he explained that he was recruited by the Ukrainian special services".