Equality Virginia, the American Civil Liberties Union and other groups are seeking the suspension of the rules to allow further public comment.
Under the state's Administrative Process Act, the board must reopen the comment period if at least 25 people make such a request. The groups say they've gathered more than 25 requests.
In April the board rejected the proposal to ban adoption agencies from discriminating against prospective parents based on sexual orientation and other issues. Attorney General Ken Cuccinelli advised that the board lacked the authority to offer such protections.