England players were booed when they took a knee before the match in a protest against racism while Raheem Sterling was pelted with cups and bottles when he scored.
A flare was thrown onto the pitch when England went 3-0 ahead and Jude Bellingham also reportedly suffered abuse.
British Prime Minister Boris Johnson called on FIFA to take action over the "disgraceful" racist abuse.
"It is completely unacceptable that England players were racially abused in Hungary last night," Johnson said on Twitter.
He urged FIFA to take "strong action against those responsible to ensure that this kind of disgraceful behaviour is eradicated from the game for good."
Harry Maguire also took to Twitter to condemn what had happened.
"Since last night's match I've spoken to my team mates and seen some of the footage. Any discrimination is totally unacceptable and the authorities must look into it," he said.
"Racism has no place in our game or society."
Since last night's match I've spoken to my team mates and seen some of the footage. Any discrimination is totally unacceptable and the authorities must look into it. Racism has no place in our game or society. pic.twitter.com/tbmDFuUpsf— Harry Maguire (@HarryMaguire93) September 3, 2021
UEFA had previously ordered Hungary to play three games behind closed doors after fans' discriminatory behaviour during Euro 2020.
FIFA said in a statement that it "strongly rejects any form of racism and violence.
"FIFA will take adequate actions as soon as it receives match reports concerning yesterday's Hungary-England game."
The Hungarian Football Federation (MLSZ) said any fans identified would face legal actions but it did not address the racism allegations.
England manager Gareth Southgate, who had ice thrown at him during an interview with BBC Radio 5 Live, said he hoped something was done.
"I've heard reports of racism, which I hadn't heard during the game," he said.
"Everyone knows what we stand for as a team, and that's completely unacceptable. It has been reported. Our head of security has spoken to the players and taken their statement. We will deal with it in the right channels.
"The world is modernising and although some people are stuck in their ways of thinking and their prejudices, they're going to be the dinosaurs in the end because the world is changing."
Hungarian foreign minister Peter Szijjarto posted on Facebook a video of England fans booing the Italian national anthem at the Euro 2020 final in July.
He wrote: "England players are complaining about last night's 'hostile atmosphere in Budapest.' The footage below was recorded in Wembley before the European Championship final. You can barely hear the Italian anthem over the whistling England supporters. Did they make similar comments back then???"
UEFA said that any disciplinary action over Thursday's incidents was a matter for FIFA but it repeated its own condemnation of racism.
"UEFA strongly condemns any discriminatory/racist behaviour and we will continue to lead the fight against racism and discrimination in the game," it said in a statement.
Information from Reuters contributed to this report.