The 55-year-old former central defender told the BBC the FA had a zero-tolerance attitude to racism and ministers should adopt the same policy.
Reported incidents of racist abuse in English football rose by 43 percent in 2018-19 compared to the previous season, according to campaign group Kick It Out.
“We in football are giving off a message about zero tolerance,” said Elliott, who is the FA’s inclusion advisory board chief. “The government has to be alongside.”
Manchester City and England striker Raheem Sterling has been highly praised for speaking out on racism.
Elliott said it was important that players received support, with potential benefits within football and society at large.
“There has to be a duty of care because the by-product of speaking out will be the positive impact and the positive behaviour in stadiums, which will then have a domino effect on societal behaviour,” said Elliott.
“With the utmost respect, this is one area where there has to be a united front — we must be together.”
The Department for Digital, Culture, Media and Sport, in a statement carried by the BBC, said it was committed to working with football authorities in tackling racism.
“Racism or any form of discrimination has no place in football or society, and we must confront this vile behaviour,” the statement said.
“We are completely committed to working closely with football on tackling racism.”
The department said it was monitoring the efforts of football authorities to tackle the issue.
