Ukrainian defence sources have accused Russia's military of demanding the surrender of their forces in Crimea.
Russia's Black Sea Fleet chief Aleksander Vitko threatened a full assault if they did not surrender by dawn on Tuesday, the sources said.
However, Interfax news agency later quoted a Russian spokesman denying that any ultimatum had been issued.
The EU and US are threatening to hit Moscow with sanctions and travel bans over its intervention in Crimea.
Meanwhile, Russia's UN envoy has said ousted Ukrainian President Viktor Yanukovych has asked Russia to use military force in Ukraine.
Vitaly Churkin said in a speech at the UN that Mr Yanukovych had made the request in writing to Russian President Vladimir Putin.
Countries have frequently attempted to justify interventions on the basis that a government has requested help to put down a rebellion.
The trouble began last month when Mr Yanukovych was ousted following months of street protests.
Since then, Russia has sent thousands of troops into Crimea, surrounding military bases and taking control of airports.
The Kremlin says its troops are needed to protect civilians in the region, many of whom are Russian speakers and have welcomed Russia's intervention.
Moscow, which does not recognise the new government, says people in Crimea have come under threat from "ultra-nationalists" since the revolution in Kiev.
Ukraine has ordered full mobilisation to counter the intervention.
No shots have yet been fired in the region, but Russia's apparent threats have intensified the rhetoric.
Reports said Ukrainian military chiefs at various bases had been told to leave their bases by Tuesday morning.
The commanders of two warships similarly told Ukrainian TV they had been given even tighter deadlines to surrender, but they had vowed instead to fight for Ukraine.
US President Barack Obama has once again accused Russia of violating international law and said Moscow was "on the wrong side of history".
American officials say they are planning to target Russian individuals and organisations with economic sanctions.
They have also once again urged Moscow to withdraw troops from Crimea, and have proposed sending international monitors to Ukraine.
The EU is also preparing to hit Moscow with sanctions.
The BBC's Chris Morris in Brussels says one official has told him the EU may even try to have Russia thrown out of the forthcoming football World Cup.
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.
Ukraine's Prime Minister Arseniy Yatsenyuk said earlier that any attempt to seize Crimea would fail, urging allies to give economic and political support to his government.
In other developments:
- Russia's rouble has fallen to an all-time low against both the US dollar and the euro
- EU leaders will meet on Thursday for an extraordinary meeting on Ukraine
- Nato will hold an emergency meeting on the crisis on Tuesday, its second such gathering in 48 hours.