Bernanke is expected to step down at the end of the year.
Appearing Sunday on ABC's "This Week," Lew said declined to discuss White House deliberations on choosing a successor.
"I'm going to keep private any conversations that we're having with the president on the question of -- of when and what kind of succession there should be," Lew said. "I think that those conversations are best left in the privacy of the Oval Office."
Speculation about a successor to Bernanke has centered on former U.S. Treasury Secretary Lawrence Summers and Janet Yellen, vice chairman of the Fed board of governors.
Summers, who was treasury secretary from 1999-2001, also served as director of the National Economic Council under President Barack Obama and was president of Harvard University from 2001-06, before resigning following criticism of comments he made in a speech questioning women's "aptitude" for science and engineering.
Yellen formerly served as president and chief executive officer of the Federal Reserve Bank of San Francisco and headed the White House Council of Economic Advisers under former President Bill Clinton.
A group of Democratic senators urged Obama last week to appoint Yellen.