France is already under a state of emergency following last year’s Islamist-claimed attacks in Paris.
The near-simultaneous assaults on a stadium, concert hall, bars and restaurants left 130 people dead and many more wounded.
In March, 32 people died in neighbouring Belgium when suicide blasts hit Brussels airport and a metro station. So-called Islamic State said it was behind both the Paris and Brussels attacks.
The event is being hosted from 10 June to 10 July at various venues.
The US has warned that the Euro 2016 football championship being held in France next month could be a target of militant attacks.
“The large number of tourists visiting Europe in the summer months will present greater targets for terrorists,” the State Department said.
Up to a million foreign fans are expected in France for the tournament, which involves the continent’s top national teams.
In its travel alert for US citizens, the State Department warns of the dangers of militant attacks throughout Europe.