He told reporters that warnings of an imminent invasion were putting Ukraine’s economy at risk.
On Thursday, US President Joe Biden said he believed Russia could attack its neighbour next month.
Russia, however, denies it is planning to invade and on Friday its foreign minister said Moscow did not want war.
While Russia has about 100,000 troops on Ukraine’s borders, Mr Zelensky said he did not see a greater threat now than during a similar massing of troops last spring.
“There are signals even from respected leaders of states, they just say that tomorrow there will be war. This is panic – how much does it cost for our state?” he told the press conference in Kyiv.
The “destabilisation of the situation inside the country” was the biggest threat to Ukraine, he said.
On Friday, US President Joe Biden said he would send a small number of troops to Eastern Europe in the “near term”, to strengthen the Nato presence in the region. He did not specify where they would be stationed or when they would arrive.
Earlier this week, the Pentagon said there were 8,500 combat-ready troops on alert, ready to be deployed at short notice.
The US has rejected a key Moscow demand that Nato rule out Ukraine joining the defence alliance – but insisted it was offering Russia a ” serious diplomatic path”.
Russian President Vladimir Putin later accused the West of ignoring Russia’s security concerns.
But he said he would study the US response before deciding what to do, according to a Kremlin readout of a call between Mr Putin and his French counterpart.
France said the two leaders had agreed on the need to de-escalate and its President Emmanuel Macron told Mr Putin that Russia must respect the sovereignty of its neighbouring states.
‘No decision made’
On Friday, US Defence Secretary Lloyd Austin said Russia had amassed enough military capacity to attack Ukraine.
He said the US was committed to helping Ukraine defend itself, including by providing more weaponry.
“Conflict is not inevitable. There is still time and space for diplomacy,” Mr Austin said.
Meanwhile the head of Germany’s foreign intelligence service said Russia was prepared to attack Ukraine, but had not yet decided whether to do so.
“I believe that the decision to attack has not yet been made,” Bruno Kahl told Reuters.
Russia last month made wide-ranging security demands from the West, including that:
Ukraine should be barred from joining Nato
Nato should end military activity in eastern Europe, pulling troops out of Poland and the Baltic republics of Estonia, Latvia and Lithuania
The alliance should not deploy missiles in countries near or bordering Russia
The US and Nato responded by saying Ukraine had the right to choose its own allies but offered Russia talks on missile placements and other issues.
If Russia were to invade Ukraine, it would not be the first time.
Russia annexed Ukraine’s southern Crimea peninsula in 2014. It is also backing rebels who seized large swathes of the eastern Donbas region soon afterwards, and some 14,000 people have died in fighting there.
‘Ukraine is not the Titanic’
By Sarah Rainsford, Eastern Europe Correspondent
This was a slightly surreal encounter. One after another, journalists asked Ukraine’s president about the threat. But Volodymyr Zelensky batted away the questions, accusing the press itself of causing panic.
On the other hand, he wasn’t contradicting the US intelligence: “I can see the 100,000 soldiers,” he eventually clarified. But he went from hinting that Russia was simply scaremongering, getting a “sado-masochistic” pleasure from seeing Kyiv sweat, to admitting that Ukraine was preparing for the possibility of all-out war.
Still, Mr Zelensky reminded people that his country has lived with the threat of Russian aggression for years – it goes in cycles – and despite the unusual size of the current deployment, he seemed determined to play down the danger.
When it came to the evacuation of some staff by some embassies, Ukraine’s leader was openly peeved: “Diplomats are like captains,” Mr Zelensky said. “They should be the last to leave a sinking ship. And Ukraine is not the Titanic.”