Zelenskiy, who has tended to play down suggestions that an attack is imminent, did not say who had suggested the date of February 16.
However, several US news organisations reported last week that Washington believed that was the date when Russian forces would be ready if Putin gave the order to invade.
"They tell us February 16 will be the day of the attack. We will make it a day of unity," he said in a video address to the nation.
"They are trying to frighten us by yet again naming a date for the start of military action."
An order had been signed to hang out national flags and wear yellow and blue banners on that day, he said.
Russia suggested on Monday it was ready to keep talking to the West to try to defuse the security crisis, while the United States said Moscow was adding to its military capabilities by the day for a potential attack on Ukraine.
Russia has more than 100,000 troops massed near the border of Ukraine. It denies Western accusations that it is planning an invasion but says it could take unspecified "military-technical" action unless a range of demands are met, including barring Kyiv from ever joining the NATO alliance.
In a televised exchange, President Vladimir Putin was shown asking his foreign minister, Sergei Lavrov, whether there was a chance of an agreement to address Russia's security concerns, or whether it was just being dragged into tortuous negotiations.
Lavrov replied: "We have already warned more than once that we will not allow endless negotiations on questions that demand a solution today."
But he added: "It seems to me that our possibilities are far from exhausted ... At this stage, I would suggest continuing and building them up."
Washington has said Russia could invade Ukraine "any day now". Putin is adding more military force and capability near Ukraine's border with each passing day, Pentagon spokesperson John Kirby told MSNBC in an interview on Monday.
"This is a military that continues to grow stronger, continues to grow more ready. They're exercising, so we believe that he has a lot of capabilities and options available to him should he want to use military force," Kirby said.
Western countries have already promised sanctions on an unprecedented scale if Russia does invade. The Group of Seven large economies (G7) warned of "economic and financial sanctions which will have massive and immediate consequences on the Russian economy".
UN Secretary-General Antonio Guterres spoke separately on Monday with the foreign ministers of Russia and Ukraine, and still believed "from his own analysis, his own hopes" that there would not be a conflict, a spokesperson said.
Moscow says Ukraine's quest to join the Western military alliance poses a threat. While NATO has no immediate plans to admit Ukraine, Western countries say they cannot negotiate over a sovereign country's right to form alliances.
Ukraine is already suffering economic damage from the stand-off as credit markets consider the probability of Kyiv defaulting.
US national security adviser Jake Sullivan told congressional leaders on Monday that Washington was considering offering Ukraine up to $US1 billion ($A1.4 billion) in sovereign loan guarantees to calm markets.
Ukraine International Airlines, Ukraine's biggest airline, said its insurers had terminated cover for at least some of its aircraft on flights in Ukrainian airspace.
German Chancellor Olaf Scholz held talks in Kyiv with Zelenskiy. On Tuesday, Scholz is due to fly to Moscow, the latest Western official to make the trip after French President Emmanuel Macron and two British ministers went last week.
Scholz said he saw "no reasonable justification" for Russia's military activity on Ukraine's border, and that "we are ready for a serious dialogue with Russia on European security issues".
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