Brazil forward Vinicius was racially abused by fans when Madrid played Valencia in May, the 10th such incident involving him that LaLiga has reported to prosecutors this season.
Infantino told Reuters that as well as applying disciplinary and competition punishments, FIFA will also pursue legal action against abusers in countries where racism happens in football matches.
"There will be no more football with racism in it. The games should be stopped immediately when it happens. Enough is enough," Infantino said on Thursday, after meeting Vinicius and the Brazil national team, who play Guinea in Barcelona on Saturday as part of an anti-racism campaign led by the Brazilian FA (CBF).
"I asked Vinicius to lead this group of players that will present stricter punishments against racism that will later be implemented by all football authorities around the world.
"We need to hear the players and what they need to work in a safer environment. We are very serious about it.
"We need harder punishments. We can't tolerate any more racism in football. As president of FIFA, I feel I needed to talk personally with Vinicius about it."
COMBAT RACISM
LaLiga and the Spanish FA (RFEF), the country's top football authorities, are under pressure to do more to combat racism after Vinicius received support from across the world following the abuse he suffered in Valencia.
"I'm overwhelmed and thankful for the huge support I received from everywhere," Vinicius said in a short statement after Brazil's training session.
"It gives me even more strength to keep fighting against racism so black kids that come after me, who maybe don't have the strength and resiliency that I have, don't need to endure the abuse I suffered.
"Infantino came today to show his support and asked me for advice on which players I think should be on the committee FIFA is creating. I think it's important and necessary, I embrace it."
Infantino, however, stressed that measures should be taken globally to eradicate discrimination in football and that political authorities and local prosecutors in every country also need to tackle racism with more determination.
"Unfortunately, racism is not a problem that exists only in Spain. It exists in many, many other countries. That is why we need to take collective measures to fight against it all over the world," Infantino said.
"We need the governments to understand that they need to educate children in school (about racism) because it's a social problem, but it's a football problem too.
"We need to stop the games, apply disciplinary and competitive measures and all of it needs to be coordinated with local authorities and governments.
"An act of racism is a criminal act... From now on, FIFA will take legal actions with local authorities too, we will join complaints filed in local justice.
"It's impossible to tolerate it anymore. We need the help and support from the authorities because the racists need to be identified, detained and banned for life from any stadium in the world.
"And it's not only in stadiums, in social media too. It's also very serious and needs the same punishment."
