Ukrainian troops have been placed on alert along the de facto border with Crimea and facing rebels in the east.
The heightened tensions follow Russian accusations that Ukraine is carrying out armed incursions into Crimea.
Ukraine's envoy to the UN asked Russia to provide evidence of the claims, and said some 40,000 Russian troops were massed on the Crimea-Ukraine border.
His Russian counterpart told the UN Security Council of Moscow's "concern and outrage" at the alleged incursions.
Crimea was annexed by Russia from Ukraine in 2014 after an unrecognised referendum.
Ukraine's SBU security service said on Thursday night that the alert level was being raised to red because of "the escalation at the administrative border with Crimea" as well as in the eastern regions of Luhansk and Donetsk.
Sporadic clashes continue in the east between pro-Russian separatists and Ukrainian forces despite a fragile ceasefire.
Russia and Ukraine accuse each other of destabilising the situation as Ukraine prepares to celebrate the 25th anniversary of independence on 24 August, and Russia prepares for a parliamentary election on 18 September.
Meanwhile, Russian state TV has broadcast an apparent confession by a man named as Yevhen Panov, saying he was part of a Ukrainian defence ministry force sent into Crimea "to carry out acts of sabotage". He said the group also included officers of Ukraine's military intelligence.
Reports in Ukraine say Mr Panov was a former volunteer fighter who has more recently been associated with a charitable organisation.
In other developments:
- The US state department has urged both sides to tone down their rhetoric, saying it has "the ability to escalate what is already a very tense situation"
- Crimean leader Sergei Aksyonov has accused the US of being behind the alleged armed incursions, saying "Ukrainian officials wouldn't have had the courage for such actions"
- Russian state-run TV reported that more than 2,500 Russian servicemen were currently taking part in a large-scale military exercise on the Black Sea
- Russia has announced its two newest ships in the Black Sea Fleet - armed with cruise missiles - will begin exercises in the eastern part of the Mediterranean from 15 August
At a meeting of the UN Security Council in New York, the ambassadors for Ukraine and Russia traded accusations.
Ukraine demanded proof of its supposed incursions from Russia, while Russia said Ukraine should be bringing an end to the conflict in the east.
Both envoys told the UN they hoped that tensions would not escalate further.
Russia's FSB intelligence agency on Wednesday said it had smashed a Ukrainian military intelligence network attempting incursions into Crimea over the weekend. "Heavy fire" from Ukrainian territory resulted in the death of a Russian soldier. An FSB employee died while trying to detain infiltrators, it said.
Russian President Vladimir Putin accused the Ukrainian government in Kiev of "practising terror" and vowed to "not let such things slide by".
Ukrainian President Petro Poroshenko described the accusations as "preposterous", calling them a "pretext" from more threats.
More than 9,500 people have been killed in the conflict in eastern Ukraine that followed the Russian annexation of Crimea, according to UN figures.
The annexation, which was itself largely bloodless, followed a vote to secede at a referendum on self-determination on 16 March 2014 - a ballot which was not recognised internationally.
The US and European Union imposed sanctions on Moscow in response to its seizure of the territory.