It is "massively important" the incoming Football Association chair and players' union chief "value inclusion", says new Kick It Out boss Tony Burnett.
Former FA chairman Greg Clarke resigned in November after "unacceptable comments" about black players.
Professional Footballers' Association chief executive Gordon Taylor will leave his role after 40 years, with both jobs set to be filled this spring.
Burnett joined the anti-racism charity as chief executive last month.
He said there are qualified candidates from minority backgrounds, but the former assistant director for diversity and inclusion at West Midlands Police believes the appointments are "less about representation".
"I'd absolutely love to see a black person or a person from an ethnic minority background in one of those roles because they came through a process which is open, fair and transparent and are the best person for the job," he told BBC Sport.
"But irrespective of the person's background, have they got the right values and experience to drive inclusion forward in the way that we need to as a game?
"In 2021, leaders of organisations, especially organisations in important positions such as [the FA and PFA], should not be appointed if they haven't got a pedigree in that field."
The FA chair and PFA chief executive roles have only ever been held by white men.
In October, the FA launched a football diversity code where clubs and organisations would have to meet a recruitment diversity target of 15% in new executive jobs.
One of those who helped develop the code was former Chelsea defender Paul Elliott, who is also the head of the FA's inclusion advisory board and has been tipped to replace Clarke.
Burnett, who played semi-professional football in his youth, hopes that appointing a black person or someone from an ethnic minority in either of the two positions would help highlight the "myth of meritocracy" which currently exists.
"There are more people called Dave than there are female chairs of our top 100 organisations," he said. "There's no way that's meritocratic."
He also wants to challenge the notion that a black person in a senior role is "representing the whole of the black community".
He added: "This applies to black players, black managers and black leaders across the world of football. When they're appointed it's because they are the best person for the job, and they happen to be black.
"They're not representing the whole of the black community. It can become 'look, we gave that black person a job and they didn't do it very well. Therefore, black people aren't very good in those types of roles'.
"So, again, it's another myth we've just got to get rid of."
Burnett, who helped launch Kick It Out's new three-year strategy last week, said that players would continue to be a key voice in the fight against discrimination, and the organisation would be appointing a players' advisory board to help in its mission.
It comes as AC Milan forward Zlatan Ibrahimovic told NBA basketball star LeBron James to stay out of politics and stick to sport after speaking about racial injustice.
Burnett said: "You could argue pretty much everything we do in life is political," Burnett added. "I think footballers now are in a really difficult situation because they are cultural icons in a lot of cases now.
"If we're asking people to be in the public eye and be representative and be role models for young people then absolutely they've got a right to have a voice about issues they care about. And we should respect that voice."
- Supporting your immune system: Can eating certain foods help?
- Radio 1's Screen Time: Daniel Radcliffe on his tense new prison-break thriller