Netherlands captain Mark van Bommel claimed in an interview published in De Telegraaf newspaper that the Oranje were forced to train away from the 25,000 fans who had turned up to watch them in the southern city of Krakow on Wednesday.
"We all heard the monkey chants," the AC Milan midfielder was quoted as saying. "We can't accept that. We reacted well and the situation was sorted.
"During the tournament, if any one of us is confronted with such a thing, we'll immediately go to the referee to ask him to intervene."
Van Bommel's comments come despite a senior UEFA official previously denying that there had been any racist chants and an angry response to a BBC programme last month which claimed that far-right gangs were rife in Polish and Ukrainian football.
The documentary showed footage of some supporters making Nazi salutes and monkey chants at black players, as well as anti-Semitism and a violent assault on a group of Asian students.
Poland, Ukraine and UEFA president Michel Platini have all tried to play down fears of racist incidents, which prompted the families of two black England players to say they would not be travelling to watch the tournament.
"It's easy to point the finger at Ukraine and Poland for racism but all countries are faced with the same problem," Platini told reporters in Warsaw on Wednesday. "We've done a lot of things but it's not a football problem, it's a social problem."
The fresh claims risk casting a cloud over the build-up to the showpiece opener of the 16-team tournament at the newly-built National Stadium in Warsaw, the first of 31 games in eight venues in the two eastern European nations.
The match kicks off at 1600 GMT and is followed by the second Group A match between the Czech Republic and Russia in the western Polish city of Wroclaw at 1845 GMT.
Van Bommel's Dutch side, who are among the favourites to challenge holders Spain and beaten 2008 finalists Germany for the title, play Denmark in the Ukrainian city of Kharkiv on Saturday.
Germany on Friday flew to Lviv, in western Ukraine, for their match on Saturday with Portugal, with captain Philipp Lahm insisting the team must prove they have what it takes to win a major tournament.
"The tension is building slowly, we have been working for two years for this moment. It's time to get going. We have a great squad, we just need to show what we can do," he said.
Euro 2012 is the first time that European football's most prestigious international tournament has been held behind the former Iron Curtain and Platini has admitted it has been a rocky road.
UEFA's executive committee awarded the tournament to Poland and Ukraine in Cardiff in 2007 in preference to Italy to widespread surprise, with even the then newly-elected Platini against the joint bid and unsure that the hosts could keep their promises in the next five years.
Events off the pitch -- from sky-high hotel prices, unfinished roads or worse, fears of racist violence and political upheaval like the jailing of Ukrainian opposition leader Yulia Tymoshenko -- have hung heavy over the tournament.
Kiev's alleged mistreatment and imprisonment of the 2004 "Orange Revolution" leader Tymoshenko has prompted a number of EU countries, including Britain, to say they will boycott matches being held in Ukraine.
Platini tried to switch the focus to the beautiful game when he addressed a news conference in Warsaw this week, insisting the tournament would be "not far off perfection" by the time the first ball is kicked.