Speaking at a news conference in Kiev on Friday, the President said he had received intelligence information, which included audio, indicating the coup is planned for December 1 or 2.
Zelensky said there was audio of Ukrainian and Russian plotters discussing the plan. Ukraine's president said the alleged plotters also mentioned the name of one of Ukraine's richest men, Rinat Akhmetov.
Zelensky alleged Akhmetov -- the owner of Ukrainian financial and industrial holding company System Capital Management (SCM) -- was "drawn into the war against the state of Ukraine" by people who surrounded him, but he didn't explain what he meant or provide any evidence to support his allegations.
Zelensky said he had invited Akhmetov to his office to speak about the alleged plot.
The Ukrainian president noted unlike his predecessors Viktor Yanukovich and Petro Poroshenko, he "will not run anywhere," but will invite all the people involved to work this out at his office.
Zelensky stressed it was an attempt to destabilize the country. "I don't think that society wants that. The country does not want that. The people who want that are those who have lost power," Zelensky said.
Akhmetov dismissed Zelensky's comments as an "absolute lie."
In a statement to CNN shared by his company, SCM, Akhmetov said he was "outraged by the spread of this lie."
"The information made public by Volodymyr Zelensky about attempts to draw me into some kind of coup is an absolute lie. I am outraged by the spread of this lie, no matter what the President's motives are.
"As a Ukrainian citizen, the country's biggest investor, taxpayer, and employer, I will continue to defend a free Ukraine, a free economy, democracy, and freedom of speech. I will do everything I can to prevent authoritarianism and censorship in Ukraine," he added.
US President Joe Biden said Friday he was "concerned" by what Zelensky had said about a planned coup.
Asked about the reports, Biden said: "I am concerned. Look we support Ukraine's territorial integrity. We support Ukraine's ability to govern itself."
Meanwhile, the Kremlin rejected Zelensky's claim Russia was involved in plotting a coup in Ukraine. "Russia never had any plans to take part. In general, Russia is never engaged in that kind of thing," Kremlin spokesman Dmitry Peskov said on a conference call with journalists Friday.
For weeks, Western officials have been expressing growing concern about Moscow's military activity on the Ukrainian border -- with the US considering sanctions if Russia invades Ukraine.
"We don't know what President [Vladimir] Putin's intentions are, but we do know what's happened in the past," US Secretary of State Anthony Blinken said during a trip to Senegal last week, referring to Russian military incursions, acknowledged or not, into neighboring countries.
The head of Ukraine's Defense Intelligence Agency, Brig. Gen. Kyrylo Budanov, told the US defense newspaper Military Times Russia was increasing troop levels and weapons systems in Crimea for an offensive which could come at any time.
The Kremlin has repeatedly said Russia has no plans to invade the Ukraine, accusing Western countries of "hysteria."