The month-long European Championship finals start in Paris on June 10, drawing the best international teams from the continent and hundreds of thousands of fans.
State Department spokesman John Kirby said the intent of the warning was not to dissuade Americans from traveling to Europe, but simply to remind them to stay vigilant.
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"I'm not aware of any specific, credible terrorist threat around these events," he said.
"This alert was issued, just as they always are, based on a cumulation of information and what we know continues to be an interest of terrorist groups to attack Western targets, and Americans specifically."
According to the warning, the threat concerns not just the French stadiums hosting matches themselves, but also areas across the continent where fans will gather in numbers.
"Euro Cup stadiums, fan zones, and unaffiliated entertainment venues broadcasting the tournaments in France and across Europe represent potential targets for terrorists," it said.
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The warning was contained in an earlier than planned update to its long-standing warning to US travelers to beware extremist attacks on transport and public gatherings in Europe.
"We are alerting US citizens to the risk of potential terrorist attacks throughout Europe, targeting major events, tourist sites, restaurants, commercial centers and transportation," it said.
"The large number of tourists visiting Europe in the summer months will present greater targets for terrorists planning attacks in public locations, especially at large events."
The French government has ruled out cancelling the prestigious tournament, which is expected to draw two million fans to cities across the country.
To strengthen security at Euro 2016, France has extended a state of emergency put in place after attacks last November on a concert hall, cafes, police and a football crowd.
Last month, French domestic intelligence chief Patrick Calvar told lawmakers that France is "clearly targeted" by the Islamic State group.
France feared a "new form of attack" in which terrorists would place bombs in areas where big crowds gather, he warned.
The updated State Department travel warning also noted that huge crowds and extra security should be expected in Krakow, Poland, during the Catholic Church's World Youth Day event between July 26 and 31.
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