It said there was a "high threat from terrorism", with stadiums, fan zones and transport hubs possible targets.
UK counter-terrorism officers have been "a key part" of security planning and a team is in France, police said.
Officers will also be stationed at Eurostar terminals and on some cross-Channel trains.
It comes after officials in Ukraine arrested a Frenchman apparently inspired by extremist right-wing beliefs to plot a series of attacks.
In his car he was carrying two rocket launchers, explosives, detonators, five Kalashnikovs and 5,000 rounds of ammunition, intelligence officials said.
The advice from the Foreign Office was updated on Monday, although officials said this was not linked to the Ukrainian arrest.
'Concern' over Euro 2016 terror threat
Move 'to shut Paris Euro 2016 fan zone'
Commander Dean Haydon, the head of Scotland Yard's anti-terror squad, said they were working with French police and security and intelligence agencies across the world to establish whether there was a threat to the Euros.
"We have seen propaganda, Islamic State (IS) and potential talk of plots against the Euros, but we're working with all those various different agencies to police and make the Euros a safe event and a safe games," he said.
"Although the ultimate policing of the event and the planning is for the French, we are all involved in properly trying to understand the threat, if there is one."
French ambassador to to the UK, Sylvie Bermann, said her country was well-prepared.
Last November, 130 people were killed in Paris when gunmen and suicide bombers linked to IS attacked the Bataclan concert hall, restaurants and bars, and the Stade de France.
Three explosions occurred outside the Stade de France stadium on the northern fringe of Paris where France were playing Germany in an international football friendly. One bomber and a passer-by were killed.
Earlier in the year, two gunmen killed 12 people in an attack on the office of the satirical magazine Charlie Hebdo. A further five people were killed when a third attacker took a Jewish supermarket hostage.
Matches at Euro 2016, which begins on Friday, will be played across several stadiums including in Bordeaux, Lille and Toulouse.
There are also a string of fan zones for thousands of supporters to watch the matches - including several in London.
England, Wales and Northern Ireland have all qualified for the tournament, with thousands of fans expected to travel to France for matches.
The England Supporters Club told the BBC the warning to fans would not stop one single fan from going to the tournament.
Fans had been waiting for two years for the Euros - this threat was not going to stop them, it said.
But Gavin Montgomery, from Guildford, Surrey, who has tickets for a match in the knock out phase of the tournament, said he was contemplating not going.
"The threats have ruined it for me. I will not be able to relax in the stadium and I definitely won't be wearing my England shirt as I fear this will attract unnecessary attention," he said.
Some told the BBC they had decided to take precautions, such as staying out of town or avoiding crowded bards.
France's 10 Euro 2016 stadiums
Euro 2016 - Uefa website
However, other fans told the BBC they remained undeterred.
Phil Padgham, from Derby, who is travelling to Marseille to watch the first England match, said: "It shouldn't stop anyone from carrying on as normal.
"Any suggestions not to travel, or that the tournament should be postponed are ridiculous, we can't be intimidated."
A police operation targeting 29 UK ports in the run-up to the tournament began on Monday, with trained spotters trying to detect football hooligans attempting to sneak out of the country.
Currently 1,841 England fans and 86 Wales supporters are subject to banning orders that stop them travelling to the tournament.
Downing Street said the French authorities had "an extensive security operation in place, with 77,000 police and gendarmes, 10,000 military personnel and a similar amount of security guards".
The UK was providing extra support including police on trains into France and more UK Border Force outbound checks, she added.
Football fans travelling to France for Euro 2016 should be "vigilant at all times" in the face of terrorism threats, the Foreign Office has said.