Downing Street warns football authorities they must 'step up efforts to tackle racism'
The Prime Minister's official spokesman welcomed the inquiry but said it was clear more needed to be done within the sport to address the issue.
He said the Government would be monitoring the response of the football authorities and was ready to take further steps "if required".
The warning came after the Professional Footballers' Association (PFA) issued a call for a Government inquiry into "racism and the rise in hate crime within football".
The PM's spokesman said: "Racism of any kind has no place in football or anywhere else and we must confront this vile behaviour.
"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."
Home Secretary Priti Patel said anyone found guilty of a racially aggravated offence under the Public Order Act could face up to six months in prison and a 10-year ban from attending football matches.
"We fully expect the entire football community to work with the authorities to bring the perpetrators to justice so we can stamp this evil out of our national game," she said.
Sports minister Nigel Adams, who was due to speak with the Tottenham directors on Monday, described the incident as "depressing".
"There is no place for racism or any kind of discrimination in football or anywhere else," he said.
European football's anti-racism protocol was implemented for the first time in a Premier League match with a statement read out three time over the public address system stating that "racist behaviour among spectators is interfering with the game".
Tottenham said it would take the "strongest possible action" against any fans found to have engaged in abuse, including banning them from the club's newly-built £1 billion stadium.
"Any form of racism is completely unacceptable and will not be tolerated in our stadium," a club spokesman said.
Bobby Barnes, the PFA's deputy chief executive, said there should be "much tougher sentencing" for those who racially abuse others.
He said racism is not specifically a football problem but a "societal problem", adding football is happy to take responsibility but law enforcement agencies also have a role.
Mr Barnes told BBC Radio 4's Today programme: "What these events are, they're actually criminal offences, they're actually hate crimes and I'd certainly call on much tougher sentencing for those found to be guilty of these particular offences."