The warning was issued a day after Ukraine announced the arrest of a suspected far-right French extremist with a massive weapons cache who was allegedly planning to attack the tournament.
The Foreign Office said there was a "high threat from terrorism" during the month-long football championship opening on Friday.
"During Euro 2016, stadiums, fan zones, venues broadcasting the tournament and transport hubs and links represent potential targets for terrorist attacks," it said.
But just three days before Euro 2016 kicks off, the threat of continued disruption to rail travel in France eased as state-run operator SNCF made an improved offer on pay and conditions to unions after all-night talks.
"There comes a time when you need to know when to stop a strike," President Francois Hollande said after a week of stoppages across the country.
"It is high time this strike ended," Prime Minister Manuel Valls added. All now depends on the CGT, France's biggest union whose leader Philippe Martinez has emerged as the chief opponent to the Socialist government's labour law reforms which have sparked three months of demonstrations and strikes.
Even if trains return to normal -- and all but eight percent of the workforce was back at work on Tuesday -- the threat of a strike by Air France pilots still hangs over the four-yearly gathering of Europe's top football nations.
The pilots are set to ground planes for four days from Saturday, just when hundreds of thousands of football supporters will be heading for France. Hollande has warned the unions they will receive little sympathy if they disrupt the tournament.
CGT activists blocked access to two of the terminals at Paris' main airport, Charles de Gaulle, for over an hour on Tuesday in an unannounced protest to call for the labour legislation to be withdrawn. The unions have also called for a national day of demonstrations on June 14.
With the threat of a jihadist attack already hanging over Euro 2016, the arrest of a Frenchman with an arsenal of weapons in Ukraine has raised new security fears. Ukraine's security service SBU said Monday the 25-year-old man, identified in France as Gregoire Moutaux, was planning to attack a string of locations in his home country.
SBU chief Vasyl Grytsak said the suspect intended to blow up "a Muslim mosque, a Jewish synagogue, tax collection organisations, transportation checkpoints and numerous other locations".
But France has made no official comment on the arrest and anti-terrorist prosecutors have not been assigned to the case, suggesting French authorities do not believe there was any imminent threat to Euro 2016.