The US State Department is warning American citizens of possible military clashes along the Russian-Ukrainian border and potential anti-American activities in Russia as Crimea votes on whether to join Russia.
It warns of the potential for escalation of tensions, military clashes, either accidental or intentional.
Moscow has shipped more troops into Crimea and repeated its threat to invade other parts of Ukraine in response to violence in Donetsk.
Two people have been killed in clashes between pro-Ukrainian and pro-Russian activists in Ukraine's eastern city of Kharkiv.
Five people were injured overnight, as gunshots were fired. Rival groups blamed each other for the violence.
Earlier, Russia and the US failed to agree on how to resolve the crisis in Ukraine's Crimea region, ahead of a secession referendum there.
Russia vowed to respect Sunday's vote - but the US said it was illegitimate.
And the United States Secretary of State John Kerry has repeated a warning that Russia will face sanctions if the referendum proceeds.
Moscow has been tightening its military grip on Crimea - the southern autonomous republic in Ukraine - where voters are deciding on whether to re-join Russia or stay with Kiev.
- the UN Security Council is to vote ona US-drafted resolution that defines Crimea's referendum as illegal
- US Vice-President Joe Biden will travel to Poland and Lithuania early next week to discuss ways to support Ukraine's sovereignty, as well as Nato members' reciprocal defence commitments
- The Pentagon says it will keep its aircraft carrier battle group in the Mediterranean Sea for several days longer than planned because of the Ukraine crisis
- Russia has moved a column of army trucks and a number of artillery pieces into northern Crimea, eyewitnesses say
- Ukrainian border guards begin checks on trains coming from Crimea into the rest of Ukraine
The violence reportedly began on Kharkiv's Svoboda Square on Friday evening and later moved to an office of a pro-Ukrainian group in the city.
Eyewitnesses said that pro-Russian activists tried to storm the rival protesters, who had barricaded themselves in.
The witnesses said that shots had been fired and Molotov cocktails thrown in.
Kharkiv Mayor Hennadiy Kernes was later quoted by Ukrainian media as saying that two people were killed and five injured.
Meanwhile, Kharkiv Governor Ihor Baluta called the incident "a provocation".
Both rival groups blame each other for starting the clashes. A criminal investigation is now under way.
This follows Thursday night's violence in Donetsk, also in the east, where at least one person died in fighting between a pro-Russian crowd and supporters of the new government in Kiev.
Ukraine accuses Russia of using provocateurs to stoke unrest on the eastern border. Moscow denies this, vowing to protect its "compatriots" from far-right radicals.
Both the US and EU have threatened tough sanctions against Moscow.
Russia's military intervention in the Crimean peninsula - part of Russia until 1954 and host to its Black Sea fleet - followed the fall of Ukraine's pro-Moscow President Viktor Yanukovych on 22 February.