Co-hosts Poland and Ukraine have already rejected claims that racist violence is rife at their football grounds, which has led to governments of some participating nations to warn travelling fans to be on their guard.
But as the tournament prepares to kick off on Friday, van Bommel said they were forced to train away from fans who had turned up for a public training session on Wednesday.
"We all heard the monkey chants," van Bommel was quoted as saying by De Telegraaf newspaper. "We can't accept that. We reacted well and the situation was sorted."
The AC Milan midfielder added: "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."
European football's governing body UEFA had said that a small group of protesters targeted the Dutch team's training ground on Wednesday but denied the demonstrations were racist.
De Telegraaf previously reported that a handful of the 25,000 fans who turned up to watch the Oranje train had made monkey noises at the side's black players during the warm-up.
But UEFA, which has a zero tolerance policy on racism at football grounds, said it had spoken to the Dutch squad and they said they had not seen or heard anything of a racist nature, the official said.
Instead, a senior official told AFP that some Polish supporters were unhappy at European football's governing body for not making Krakow one of the four Polish host cities for the tournament and made themselves heard.
One journalist from Dutch radio RNW, however, maintained that some of the short-lived protest that was cut short by stadium officials was racist.
Photos showed some fans with banners with a red line through the Euro 2012 logo.
Coach Bert Van Marwijk, who is van Bommel's father-in-law, had said after the training session: "I congratulate the spectators who created an extraordinary atmosphere in the stadium."
The Netherlands play their opening match in Group B against Denmark in the Ukrainian city of Kharkiv on Saturday.
Last month, BBC television aired a documentary suggesting that far-right gangs used football matches in both host nations as recruiting grounds and showed footage of fans making Nazi salutes and monkey noises at black players.
The British government, among others, issued a travel warning to fans about the risk of racist violence, prompting the families of two black England players to say they would stay at home.
Former England captain Sol Campbell, who is black, publicly advised black and ethnic minority fans not to travel while Italy's Mario Balotelli even said he would walk off the pitch if he were subject to racial taunts.
Poland, Ukraine and UEFA president Michel Platini have said that racism was not just a problem for the host nations.
"Across the world and Europe in particular, you can see that over the last few years there's been more and more nationalism and you see it at a lot of matches," Platini told reporters in Warsaw on Wednesday.
"It's easy to point the finger at Ukraine and Poland for racism but all countries are faced with the same problem... We've done a lot of things but it's not a football problem, it's a social problem."
Platini also said it has measures in place to ensure that racist incidents are kept to a minimum and referees now had the power to take players off the pitch.
"We would support them if they stopped the match," he told reporters.