"There is a high threat from terrorism," the foreign ministry said in a travel advisory for supporters.
"During Euro 2016, stadiums, fan zones, venues broadcasting the tournament and transport hubs and links represent potential targets for terrorist attacks," it said.
"You should be vigilant at all times, especially in areas hosting Euro 2016 events and follow the advice of local French authorities," it added.
Ukraine said Monday it had arrested a suspected far-right French extremist who was allegedly planning to attack the Euro 2016, adding to security fears over Europe's showcase football tournament.
But Prime Minister David Cameron's spokeswoman told reporters the new advisory was "not a response to a new or specific threat or to yesterday's arrest".
"It's to provide further information on the risk of terrorism during the Euro 2016 tournament for fans. It's not a change in terms of the advice about whether or not to travel," she said.
She also said that more counter-terrorism officers would be deployed on trains to France and on Britain's borders but did not give further details.
According to the warning, the threat concerns not just the French stadiums hosting matches, but also areas across the continent where fans will gather in numbers.
The Euro 2016 runs from June 10 to July 10.
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