Play was stopped during Chelsea’s win at Tottenham on Sunday after Antonio Rudiger said he heard monkey noises.
The Professional Footballers’ Association has called for a government inquiry following the alleged abuse.
“Racism of any kind has no place in football or anywhere else and we must confront this vile behaviour,” said a Downing Street spokesperson.
“Clearly there remains more work to be done by the football authorities in tackling this issue and we are committed to working with them on this to stamp it out.
“The FA, Premier League and English Football League have significantly stepped up their efforts, but we expect them to continue to prioritise this issue and to consult with both players and supporter groups, and we will be monitoring how the football authorities implement their plans through the season.
“We will continue working with the authorities on this, including the Professional Footballers’ Association and we don’t rule out taking further steps if required.”
Cheslea centre-back Rudiger reported the alleged racist abuse from the crowd to his captain Cesar Azpilicueta, who told referee Anthony Taylor and the game was stopped.
Shortly after the stoppage, an announcement made over the public address system warned that “racist behaviour is interfering with the game”.
Second and third addresses followed with the game heading towards its conclusion.
The Metropolitan Police is working with Tottenham to identify anyone responsible, with the club promising to “take the strongest possible action”.
The FA said: “We are working with the match officials, the clubs and the relevant authorities to fully establish the facts and take the appropriate steps.”
Speaking in August, Premier League chief executive Richard Masters said the organisation was “determined to tackle discrimination”.
“One incident is one incident too many,” added Masters. “We want to encourage fans when they see other supporters making discriminative comments, abusing other people they do report it and that’s important.”
‘Idiots responsible for abuse a disgrace’ – Vertonghen
Spurs defender Jan Vertonghen says anyone responsible for racist abuse are “idiots” and do not identify with the club.
“I have got no idea how people still, or ever, thought this way,” said the Belgium international.
Vertonghen says he “loves” and the UK because its multi-cultural society and that reports of racism “hurts”.
“I didn’t hear anything, but if these things are still happening it is a disgrace and we should act strongly against it,” he added.
“Sometimes you think people are smarter than this. I am very convinced it is just a minority, but it is very wrong.
“I don’t know if it is getting worse. It shouldn’t be there in any way. I just can’t get my head around how people still do this. I have got no words for it.
“If any of their players or our players are affected then I apologise in the name of Spurs, but they are minority idiots. We don’t identify with these people.”
Premier League managers on racism
Tottenham manager Jose Mourinho: “I can repeat I am totally, totally against racism in society and obviously against racism in my football world.
“I feel very sorry every time something happened and I will always support every decision the authorities can make.
“Society needs help. And then football is a micro-society. Do we need help? Yes. But society needs help. We need to eradicate any form of discrimination and this case we are talking about racism. Football and society needs help.”
New Everton boss Carlo Ancelotti said: “It is a problem everywhere. I had a big fight last year in Italy (as Napoli manager) when Kalidou Koulibaly was abused in the stadium in Milan. We have to be strong.
“Football cannot allow people to abuse others. Every federation in the world has to be strong against this.”
Sheffield United boss Chris Wilder: “I have always thought that it’s a societal problem and is attached to football grounds.
“If you are sat next to someone who is doing it, saying terrible, terrible things, just out them. Out the people next to you. Be brave and police your own football ground. Isolate them and let’s get them out.”
Newcastle manager Steve Bruce: “I think I’m like everybody else, you’re sickened and saddened by it.
“We criticise Europe and parts of it, and unfortunately it’s creeping back into our game. Really, it’s a society thing and I don’t think it will be long before we see teams walk off the pitch. Nobody wants to see it.
“You just can’t come to terms that somebody wants to go to a football match and do that to an individual. I’m like everybody else, let’s find out whoever it was and ban them for life.”