The alert, issued by the US State Department, is the first specific warning in recent months, although it does not indicate any likely targets.
US citizens in Ukraine are being advised to leave the country if it is safe to do so.
The alert comes as Ukraine marks 31 years as an independent state, after its split from the Soviet Union.
Any celebrations have been muted, as the government in Kyiv cancelled planned public ceremonies to mark Ukraine’s independence day.
President Volodymyr Zelenskyy has vowed to fly the Ukrainian flag in parts of the country still under Russian control, six months after Moscow launched its full-scale invasion.
Russia's Vladimir Putin and Ukraine's Volodymyr Zelenskyy
Reuters/PA
So far, the trans-Atlantic alliance has managed to maintain its united front, agreeing wide-ranging sanctions against Moscow, as well as financial and military support for Kyiv.
But that consensus will be sorely tested as the West enters a bleak winter of rising food prices, dwindling supplies of energy and the likelihood of recession in many economies.
Former British Army officer Colonel Richard Kemp foresees fractures in that alliance in the months ahead.
He told GB News: “I think we’re going to see a reduction in the unity as we get into the winter.
“Particularly with the energy problems that countries like Germany will face, both domestically and in industrial terms.
“That I think will lead to Germany and other Western countries, particularly NATO countries, reducing their support for Ukraine and probably pressuring President Zelenskyy to come to terms with Russia.”
In the early days of the war, Russian tanks rolled quickly across Ukrainian soil, encircling key population centres, including Ukraine’s second city Kharkiv, which suffered many weeks of bombardment.
But a rush towards the capital was halted and then reversed, thanks to a combination of fierce Ukrainian resistance, growing stocks of sophisticated Western weapons systems and poor strategic and logistical planning by the Russians.
Angela Stent, Senior non-Resident at the Brookings Institution, said: “What we’ve seen in the past few months is the Russian have made some gains, they’ve taken some territory in the Donbass region, but they’re very incremental.
A Ukrainian serviceman jumps from a military vehicle near a frontline in Mykolaiv region
STRINGER
“The Ukrainians are pushing back, we’re back to World War 1-like trench warfare.
"It’s a war of attrition so, unless something dramatic happens to change this, this war can grind on for a long time.”
In recent weeks, the Ukrainian counter-offensive has seen their forces begin to take back control of some key areas.
They have massed troops near the strategically important city of Kherson, currently under Russian control.
Six months on from Russia’s invasion, Ukraine’s armed forces and its people are in this fight for the long haul – they desperately hope their Western allies remain just as committed.