The Consortium for Citizens with Disabilities' education task force, a coalition of more than 20 groups, said eliminating the cap would "wipe out a decade of progress."
Education watchers were disappointed last month, when a promising bipartisan-backed bill failed to make it through the lame-duck session.
Rice has close ties to Bush's family, having served as secretary of state under Jeb Bush's brother, former President George W. Bush, from 2005 to 2009.
The Obama administration wants to expand eligibility for child-care programs and bolster quality.
John Kline, R-Minn., chairman of the House education committee, said he will use the Student Success Act, the bill he ushered through the House in the 113th Congress, as the starting point for the legislative process.
The U.S. Department of Education has offered seven states an early deadline and a very special, expedited review process for renewing their No Child Left Behind waivers, but not everyone is taking it up on the offer.
The panel's priorities for a Head Start rewrite include reducing unnecessary regulatory burdens and boosting coordination between Head Start and state and local programs.
After two-and-a-half hours of hearing from a diverse panel of witnesses, chairman Lamar Alexander, R-Tenn., still hadn't come down on one side of the testing debate.
President Barack Obama used his penultimate State of the Union address to call for a dramatic expansion in college access and increased investments in early childhood.
During the NCLB reauthorization process, Sen. Chris Murphy, D-Conn., said he'll "be judging the draft and every amendment to the draft in part by whether it preserves the gains we've made for kids with learning disabilities."