President Barack Obama will officially introduce Cordray, now the agency's enforcement chief, Monday, The Columbus (Ohio) Dispatch reported.
"Richard Cordray has spent his career advocating for middle-class families," the president said in a statement released Sunday.
Sen. Sherrod Brown, D-Ohio, praised the choice and predicted Senate confirmation "unless they get to be hyper-partisan. My only fear is Republicans don't think we should have consumer protection rules."
Elizabeth Warren, who conceived and set up the bureau, was fiercely opposed by Republicans and was considered difficult to get past the Senate.
Obama thanked Warren "not only for her extraordinary work standing up the new agency over the past year, but also for her many years of impassioned leadership, and her fierce defense of a simple idea: Ordinary people deserve to be treated fairly and honestly in their financial dealings."
Warren, who hired Cordray after he was defeated for re-election in November, said: "Rich has always had my strong support because he is tough and he is smart -- and that's exactly the combination this new agency needs."
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