Bernanke, if confirmed by the Senate, would on 31 January replace Alan Greenspan, who will step down after 18 years in one of the most powerful economic posts in the United States.
Bernanke was a member of the Federal Reserve before moving to the White House this year to lead Bush's Council of Economic Advisers. He previously was the head of the economics department at Princeton University.
"He's earned a reputation for intellectual rigour and integrity," Bush said in a brief White House appearance with Bernanke on Monday.
"He commands deep respect in the global financial community. And he'll be an outstanding chairman of the Federal Reserve."
Bush acknowledged that Bernanke would have difficult shoes to fill after Greenspan's tenure.
"For nearly two decades, Chairman Greenspan has shepherded our economy through its highs and its lows," Bush said.
Greenspan is to step down after
18 years in the powerful post
"Under his steady chairmanship, the United States economy has come through a stock market crash, financial crises from Mexico to Asia, two recessions, corporate scandals, and shocks ranging from devastating natural disasters to a terrorist attack in the heart of America's financial centre."
Bush added that "Ben Bernanke is the right man to build on the record Alan Greenspan has established."
Bernanke, 51, has a reputation for political independence. He has been a proponent of specific inflation targets as used by other central banks but opposed by Greenspan.
Bernanke made his mark as a deflation fighter shortly after joining the Fed in August 2002.
In a speech in November 2002, Bernanke laid out an aggressive Fed strategy to ward off deflation.
In addition to rate cuts, Bernanke said the Fed had a variety of weapons to ward off deflation, including purchasing Treasury bonds back from the market.
Members of the Fed board serve 14-year terms and must be confirmed by the Senate. The chairman is chosen by the president from among the board members and serves a four-year term.
Experts said they do not expect Bernanke's nomination to run into trouble on Capitol Hill.