Russia's state news agencies quoted the defence minister, Sergei Shoigu, saying Putin ordered the test at 2pm Moscow time on Wednesday.
The Kremlin was unable to confirm the order.
The provocative display of Russian military muscle comes as Ukraine's interim leadership urgently sought to win the confidence of its divided and economically ravaged nation with the creation of a new unity government.
Pro-European protesters filled Kiev's Maidan square – the seat of the Ukrainian revolution – preparing to meet the replacement cabinet proposed by Oleksandr Turchynov, Ukraine's acting president, on Wednesday, as Turchynov dismissed the country's feared riot police.
Turchynov had warned on Tuesday that the country faced a serious threat from separatism amid fears the Kremlin may be stoking pro-Russian sentiment in the Crimean peninsula.
"We discussed the question of not allowing any signs of separatism and threats to Ukraine's territorial integrity and punishing people guilty of this," Turchynov said after meeting key officials.
Before Wednesday's military manouevre, a visiting Russian parliamentarian said on Tuesday that Moscow would act in the event of heightened tension over the Crimean peninsula.
"If the life and health of our compatriots is under threat, we will not stand to one side," the parliamentarian, Leonid Slutsky, said after arriving in the regional capital of Simferopol for a one-day visit.
Slutsky, who leads the Russian Duma's committee for relations with former Soviet states, declined to say what sort of action Russia might take.
AFP reported that two armoured personnel carriers were deployed near Russian military installations in Ukraine's second port of Sevastopol on Tuesday.
One of the vehicles was on a base belonging to Russia's Black Sea fleet while another was parked in the courtyard of a navy building in the city centre.
A spokesman for the fleet in Sevastopol refused to comment on the deployment of the vehicles but local media reported that they had been sent out in case of "terrorist attacks".
Sergei Lavrov, the Russian foreign minister, warned the EU and US not to try to shape Ukraine's future.
"It is dangerous and counterproductive to try to force upon Ukraine a choice on the principle 'You are either with us or against us'," Lavrov told a news conference in Moscow.
Russia and the west should use political contacts in Ukraine to calm the situation and not seek advantage when national dialogue was needed, Lavrov added.
The US secretary of state, John Kerry, and the British foreign secretary, William Hague, have rejected the suggestion that Ukraine is being forced to choose between east and west.
"This is not a zero-sum game, it is not west versus east," said Kerry after hosting Hague in Washington.
The Ukrainian government faces foreign debt payments of $13bn (£7.8bn) this year and has less than $18bn in its fast depleting coffers – a grim equation that has forced it to seek as much as $35bn from western states.
Moscow has frozen payments on a massive bailout package promised by its president, Putin, to Yanukovych as his reward for rejecting closer EU ties.
Both the United States and Britain have publicly backed the idea of putting together an economic rescue for Ukraine that would be overseen by the International Monetary Fund.
Hague stressed after the talks with Kerry that "this is a country that needs financial assistance from many sources, including from Russia".
However, the EU foreign policy chief, Catherine Ashton, wrapped up a two-day visit to Kiev on Tuesday by mentioning only a "short-term" economic solution for Ukraine while saying nothing about extending the billions of dollars in credit requested by Turchynov.
On Tuesday, the parliament in Kiev voted to send Yanukovych to The Hague to be tried over the violence that led to at least 82 deaths in Kiev last week.
He is on the run and believed to be in Crimea, but has not been seen in public since Sunday.
The parliament voted that Vitali Zakharchenko, the former interior minister, and Viktor Pshonka, the prosecutor general, should also be sent to the international criminal court.
They have fled Kiev, along with other key Yanukovych aides.
There were reports on Tuesday afternoon that Andriy Kluyev, Yanukovych's chief of staff, had been wounded during a gun battle.
He was reported to have been shot in the leg after his car came under fire when he was travelling back to Kiev, allegedly after visiting Yanukovych in Crimea.
None of the details could be confirmed.
Outside Kiev, Yanukovych's extravagant residence has been opened to the public, and Ukrainian investigative journalists have begun releasing to the web incriminating documents found at the mansion, in a project called "Yanukovych leaks".
They say they have found evidence of corruption as well as plans to clear Independence Square of protesters using force.
Pictures from Pshonka's mansion were also posted online, including one shot of a portrait of the prosecutor general dressed as a Roman emperor and another of hundreds of lavish gold ornaments.
In The Hague, the international criminal court said it had not yet received a request from the new Ukrainian government to investigate events in Kiev.
Oleh Tiahnybok, leader of the nationalist Svoboda party, said: "It is very important that we had a positive vote today. Now we are inviting all the people of goodwill who have any materials including video, photos or papers that we may need to properly submit to The Hague tribunal the papers about crimes against people, crimes against Ukrainians, and violations of human rights that were committed by those criminals in Yanukovych's regime."
Senior figures in Washington have claimed that in the days before he fled the capital, Yanukovych spent at least an hour in consultation over the phone with Joe Biden, the US vice-president.
According to the anonymous US officials, Biden found the beleaguered Ukrainian leader to be initially defiant, accusing protesters in control of the Kiev streets of terrorism.
Speaking through a translator from his office in the west wing of the White House, Biden reportedly warned Yanukovych that leaders in his position were often "a day late and a dollar short" in their attempts to appease political protesters.
The parliament will select a new prime minister on Thursday, and presidential elections are scheduled for 25 May.
The retired heavyweight boxer Vitali Klitschko confirmed on Tuesday that he would run.
Yulia Tymoshenko, the former prime minister who was released from jail on Saturday, has not indicated whether she will stand.
She said she would travel to Germany in March for treatment for a back problem which has confined her to a wheelchair.
Vladimir Putin has not yet commented on the situation, but the parliamentary visit to Crimea will further stoke tensions.
Crowds in Simferopol and Sevastopol on Tuesday called on local authorities to reject Ukraine's new government.
A crowd of about 100 people gathered outside the local administration in Sevastopol cheered when a Russian armoured personnel carrier rolled past – an apparently routine occurrence in the port city.
The night before, the city council handed power to Aleksei Chaliy, a Russian citizen, while more than 1,000 people gathered around city hall chanting.
"Russia, Russia, Russia" and "a Russian mayor for a Russian city".
Viktor Neganov, a Sevastopol-based adviser to the interior minister, condemned the events in the city as a coup.
"Chaliy represents the interests of the Kremlin, which probably gave its tacit approval," he said.
The Crimean peninsula, which is the only region of Ukraine with a majority of ethnic Russians, was Russian territory until 1954.
Moscow recently extended its lease on a large naval base in Sevastopol to 2042.
Source: The Guardian