Clarke resigned on Tuesday evening after making a series of offensive remarks during an evidence session with MPs on the Digital, Culture, Media and Sport committee, including using the word "coloured".
Peter McCormick, who chairs the FA's professional game board, has been appointed as interim chairman and will assist chief executive Mark Bullingham in the hunt for Clarke's permanent successor.
Bullingham said in a statement: "With other board directors, Peter and I are finalising the process for recruiting a new independent chair of the FA. Our aim is to have one in place by the end of the first quarter.
"Our process will be open and conform to the Diversity Code, ensuring that we are able to select the best candidate from a diverse talent pool.
"We know that football has the power to unite and bring people together. We have a clear and focused vision on how we will do that, improving opportunities within football by breaking down barriers so that we have an inclusive game that we can all be proud of.
"We absolutely recognise that there is more work to be done, but our commitment and passion to help improve lives, embrace diversity and create opportunities will remain fundamental to our organisation for years to come."
Kick It Out executive chair Sanjay Bhandari, who was scathingly critical of Clarke in a statement on Tuesday, has ruled himself out of the running for the position, but says he is happy to offer any assistance he can in identifying the right person.
"I don't think the FA is going to ask me about the role," he said. "I'm still relatively new to football, it's way too early for someone like me. But I am really happy to offer any help they need in identifying the next candidate.
"It's important we have a good relationship with the leadership of the FA and I have really good relationships with the chief executive (Mark Bullingham) and lots of other people at the FA, but I don't think they will be knocking on my door."
Paul Elliott, the chairman of the FA's inclusion advisory board, has been reported as a contender, while Bobby Barnes, the deputy chief executive of the Professional Footballers' Association also has strong credentials, as has the FA's head of women's football Baroness Sue Campbell.
Bhandari set out what he saw as the key attributes the successful candidate must have.
"On equality, diversity and inclusion there's a big role, not just because of what Greg said yesterday but because they have made it a tent-pole of their strategy," he said.
"This is part of who they are and what they want to be. Someone has to get that and be able to drive that across the game and also internally within the FA.
"It has to be someone who understands that intrinsically, they need that genuine empathy which I think is what was missing yesterday.
"They would also have to be pretty good at governance. The board is not very representative of the game - there is no representation from black players who make up such a significant constituency within the game.
"That needs to be dealt with. The governance structure of the FA as a whole is a tough nut to crack, it's the archetypal archaic old men in blazers. I know previous chairmen have tried that."
He added: "They would also need to have the networks to be creative and broker agreements between the Premier League, the EFL and the Government on the future financial structure of the game.
"And on the opportunities side, there's hopefully a World Cup bid coming (for 2030) so someone needs to have those relationships across international football to put our best foot forward."
Clarke is also a vice-president of Fifa and sits on its ruling council. The world governing body has yet to comment on what happens next, but England could lose a position of real influence within the top level of international football at a time when it is considering a World Cup bid.
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