The Kremlin claimed two drones had been used in the alleged attack but they were disabled by Russian defences.
The Russian news agencies reported that Mr Putin was not in the Kremlin at the time and there was no material damage to buildings.
Unverified footage on Russian social media showed smoke over central Moscow.
Ukraine is yet to comment on the allegation.
The Kremlin called the incident a “planned terrorist act” and said it reserved the right to respond when and how it sees fit.
“The Kremlin has assessed these actions as a planned terrorist act and an assassination attempt on the president on the eve of Victory Day, the May 9 Parade,” RIA said.