Helen Grant, the new sports minister, will meet the Football Association on Wednesday to demand answers over the lack of diversity on its flagship commission in the wake of stinging criticism from one of its own board members.

The FA was also accused on Saturday of making "a big mistake" and of a "public relations disaster" by football's two most important anti-discrimination bodies, Kick It Out and the European body Football Against Racism in Europe (Fare).

Grant, who is also equalities minister, spoke out after the FA board director Heather Rabbatts raised serious concerns about the lack of diversity among the eight all-white, all-male members of the panel announced so far and the lack of consultation over their recruitment. Rabbatts wrote a strong letter on Friday to her fellow directors accusing the FA of letting down the ethnic minority community.

"Sports governing bodies must reflect the make-up of the diverse society that we live in," said Grant, who was at Wembley when England beat Poland. "I expect the FA to ensure that voices from all backgrounds are heard loud and clear and contribute to this important piece of work to help strengthen English football. I will discuss the issue with the FA."

Rabbatts was also concerned that the FA had failed to consult the board on who should sit on the panel, which was set up to address the growing dearth of homegrown talent in the English game. The eight names so far confirmed – with an average age of 57 – were announced after the speech this month by Greg Dyke, the FA chairman, to the Leaders in Football conference.

After a week in which England qualified for the World Cup only to be trapped in the fallout from Roy Hodgson's "space monkey" team talk, the FA is faced with the familiar prospect of a looming crisis of its own making.

"This has been a disaster for the FA in public relations terms. Heather must have been at her wits' end to go public like this," said Kick It Out's chairman, Lord Ouseley. "I raised this with the FA chairman as soon as he made the announcement and he told me he was conscious it was an all-white, all-male membership and that he was trying hard to get an ethnic minority representative. I find it very surprising therefore that someone as close to the chairman as Heather Rabbatts, with all her experience and contacts, was not asked for her advice on ensuring diversity."

The Fare executive director, Piara Powar, said: "I think they have made a big mistake. There has been a big debate among black players and the ethnic minority communities about the lack of diversity on the commission. If they appoint someone on to the commission now it very much looks like tokenism. If an FA board member such as Heather Rabbatts cannot make headway with the FA on this, then who can?"

Dyke, in response to Rabbatts, said he was disappointed she had gone public with her concerns and "surprised" by her comments because they implied the FA had not considered the panel's ethnic make‑up. "I do accept we made a mistake announcing only part of the membership of the commission when we did but to suggest we never considered the ethnic balance of the commission is unfair," he said.

"We originally had Clarke Carlisle as a member but the PFA decided they would rather have their new chairman on the commission. We and we also identified other individuals from the BAME [black and minority ethnic] community who we felt would add strength and value. Unfortunately, as they are active in football on a day-to-day basis they felt the time commitments would be prohibitive.

"As you know we still want to see people with relevant experience from the BAME community on the commission and giving evidence to it." He promised the commission would engage in an "exhaustive and fully inclusive process".

The row over the make up of the commission comes almost exactly a year after English football was hit by the fallout from on-field racist incidents involving John Terry and Luis Suárez, which left some black players furious at their lack of representation in the game.

As part of a process led by the prime minister, the FA came up with a 99-point document to foster more inclusion. It included a new inclusion committee, to be chaired by Rabbatts, a former chief executive of Lambeth council and chair of Millwall FC, who was appointed to the FA board in December 2011 as one of two new nonexecutive directors designed to improve the FA's corporate governance.

In her letter Rabbatts said she had tried to raise the issue privately but there had been a "refusal to understand" her position. "I believe that the lack of proper consultation on the make-up of the commission, the fact that no approval was sought from the board, releasing the names of the 'chosen' individuals at Leaders in Football, the composition of the commission itself and the lack of diversity, have all meant that the opportunity to lead an informed debate on the future of English players has been singularly damaged," wrote Rabbatts, who was born in Jamaica and moved to Britain aged three.

"I make the comments about diversity not because they are additional to this matter but because they lie at its heart," she added. "What is required is not tokenism but the involvement of individuals who have direct and relevant experience of what it means to represent their country while coming from diverse cultural backgrounds.

"By proceeding along this current path we are not only failing to reflect our national game but we are also letting down so many black and ethnic minority people – players, ex-players, coaches and volunteers, who have so much to offer and are so often discouraged and disheartened by the attitudes they encounter. The FA should be leading by example not reinforcing entrenched attitudes."

Rabbatts is not believed to be considering her position on the FA board over the matter but is waiting to see what the governing body's response to her intervention will be. Aside from the lack of diversity, which has also been criticised by Sol Campbell, among others, the panel has also been criticised for its parochial feel and lack of overseas expertise. Gary Lineker called it "pointless".

As well as Dyke, the commission will include the former England manager Glenn Hoddle, the Football League chairman, Greg Clarke, and the FA vice-chairman Roger Burden, the League Managers Association chairman, Howard Wilkinson, the Professional Footballers' Association chairman, Ritchie Humphreys, the Crewe director of football, Dario Gradi, and the former England defender Danny Mills.

The FA says that the commission, which will take evidence from a wide range of people and organisations within the game before reporting next March, remains incomplete and "one or two" more names will be added. Dyke said in his letter that the full commission would have been unveiled already had the row over Hodgson's half-time comments not "blown up" in the way it did.