Ousted Ukrainian President Viktor Yanukovych asked Russia to send troops across the border to protect civilians, Moscow's UN envoy has claimed.
Vitaly Churkin told a Security Council meeting Mr Yanukovych wrote to Russian President Vladimir Putin on Saturday.
Thousands of Russian troops have been pouring into Ukraine's Crimea region.
Russian forces have also been holding military exercises near Ukraine's borders, but now Putin has ordered them back to base, the Kremlin says.
Ukraine said Russia had set a deadline for its forces in Crimea to surrender by 03:00 GMT. So far there have been no reports of any incidents.
Russia has denied issuing any ultimatum.
Western ambassadors dismissed Russia's arguments as groundless.
The Kremlin has established de facto military control in Crimea. There are growing fears that it might try to seize more land in eastern Ukraine, where a number of people support closer ties with Moscow.
Ukrainian officials say reports suggest that there is a Russian military build-up near Ukraine's eastern border.
In another development, the Pentagon said it was putting on hold all military-to-military engagements between the US and Russia. These include exercises, bilateral meetings, port visits and planning conferences.
However, a Pentagon spokesman said that "there has been no change to our military posture in Europe or the Mediterranean; our Navy units continue to conduct routine, previously planned operations and exercises with allies and partners in the region".
Crimea, Ukraine's autonomous republic, and several cities in south-eastern Ukraine have been rocked by instability since Yanukovych was ousted last month.
On Monday, pro-Russian protesters attempted to storm government buildings in the eastern city of Donetsk and Odessa, in the south.
Russian troops and pro-Moscow militias continued an apparent blockade of Ukrainian military bases across Crimea.
Kiev's UN envoy Yuriy Sergeyev says Russia has deployed about 16,000 troops to the peninsula.
Ukrainian defence sources accused Russia's Black Sea Fleet chief Aleksander Vitko of threatening a full-scale assault if they did not surrender by dawn on Tuesday.
A Russian spokesman was later quoted as denying that any ultimatum had been issued.
But Ukrainian military personnel besieged in their bases have been waiting nervously for the 03:00 GMT deadline, many of them preparing for an attack.
At the UN in New York, Churkin offered the Security Council a copy of the letter Yanukovych had sent.
He said the ousted leader had described Ukraine as on the brink of civil war, with civilians being persecuted simply for speaking Russian.
Ukraine's officials deny this, pointing out that an overwhelming majority of people in Crimea and well as in a number south-eastern cities in Ukraine's mainland speak Russian in their day-to-day lives.
Churkin quoted from the letter: "I would call on the president of Russia, Putin, asking him to use the armed forces of the Russian Federation to establish legitimacy, peace, law and order, stability and defending the people of Ukraine."
Churkin reiterated Moscow's view that Yanukovych is Ukraine's legitimate leader, not interim President Olexander Turchynov.
Western ambassadors refused to accept Russia's justification, continuing to accuse Moscow of violating international law.
"Russian mobilisation is a response to an imaginary threat," said US envoy Samantha Power.
"Military action cannot be justified on the basis of threats that haven't been made and aren't being carried out."
One Ukrainian navy officer said his forces were facing intense pressure from Russia to support them
UK ambassador Mark Lyall Grant also rejected Russian allegations of terrorism and threats against ethnic Russians in Ukraine.
"It is clear that these claims have simply been fabricated to justify Russian military action," he said.
He dismissed Yanukovych's letter to President Putin as meaningless.
A number of Western nations have urged Russia to withdraw its troops and accept international observers in Ukraine.
The US and EU have threatened sanctions including possible travel bans and targeted economic measures.
However, a British official was photographed holding policy documents that suggest the UK will not seek to curb trade with Russia or close London's financial centre to Russians.