The nominations, announced Friday, mark the latest move by Bush to put his imprint on the Federal Reserve.
Kroszner is an economics professor in the University of Chicago's graduate school of business and a previous member of the Council of Economic Advisers. He directs the university's George J. Stigler Center for the Study of the Economy and the State.
Kroszner also is a research associate at the National Bureau of Economic Research and a member of the board of directors of the National Association of Business Economists. He received a bachelor's degree from Brown University and a master's and doctorate from Harvard University.
Warsh is a special assistant to the president for economic policy at the White House, managing domestic finance, capital markets and banking issues. Previously, Warsh served as executive director and vice president of mergers and acquisitions in the investment banking division of Morgan Stanley.
Warsh received a bachelor's degree from Stanford University and a law degree from Harvard.
The nominations are subject to Senate approval.
The selections come as the Senate is preparing to clear White House economist Ben Bernanke to be the next chairman of the Federal Reserve. Bush nominated Bernanke, a former Fed member and Princeton economics professor, in October.
Bernanke will take over the Fed from Chairman Alan Greenspan, who retires on Tuesday after 18 years at the helm.