A Football Association spokesman said the proposals, which include an expansion of the council to bring more diversity, would now go to the annual general meeting of shareholders for final approval on May 18.
Only one woman, Heather Rabbatts, now sits on the board and the changes have been criticised by some as not going far enough.
Anti-discrimination group Kick It Out said last month that the reforms were "a sham" that still failed to give sufficient representation to minority groups.
Former FA chairman David Bernstein has said the soccer body risks missing an opportunity for real change and could end up with a "fudged compromise".
Clarke previously threatened to quit if the proposals were not approved.
British Sports Minister Tracey Crouch warned in February that the FA was in danger of losing 30 million pounds ($37.29 million) of public funding and that legislation could be brought in to force change.
The FA has proposed reducing its board from 12 to 10 members and increasing the number of women on it to three by 2018.
Nobody will be allowed to serve more than three terms of three years, and 11 new members will be added to the 120-strong FA Council "to ensure it better reflects the inclusive and diverse nature of English football".
(Reporting by Alan Baldwin; editing by Mark Heinrich)
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