Lew, 57, who also has served as director of the Office of Management and Budget, has been offered the Treasury post by Obama, according to the person, who asked for anonymity to discuss personnel matters.
Geithner, 51, the only remaining member of Obama's original economic team, has told White House officials he doesn't' want to serve in a second term and intends to leave the job by the end of the month.
Lew's nomination as Treasury secretary is subject to confirmation by the Senate.
The next Treasury secretary will play a leading role in working with Congress to raise the government's $16.4 trillion debt ceiling. The U.S. reached the statutory limit on Dec. 31, and the Treasury Department began using extraordinary measures to finance the government. It will exhaust that avenue as early as mid-February, the Congressional Budget Office says.
As a former aide to the late House Speaker Tip O'Neill, a Massachusetts Democrat, and a two-time director of the Office of Management and Budget, Lew has experience on both ends of Pennsylvania Avenue. He's spent most of his career in government, with a brief detour to Wall Street, where he worked as a managing director for Citigroup from July 2006 until joining the administration when Obama first took office.
Obama is filling out his Cabinet for a second term. On Dec. 21, he announced his choice of Senator John Kerry, a Massachusetts Democrat, as secretary of state, replacing Hillary Clinton. On Jan. 7 Obama said he would nominate former Republican Senator Chuck Hagel as defense secretary.
Obama has several other cabinet and cabinet-rank jobs to fill, including commerce secretary, Environmental Protection Agency administrator, U.S. trade representative and director of the Office of Management and Budget.
Among the leading candidates to replace Lew as Obama's chief of staff are Denis McDonough, currently a deputy national security adviser, and Ron Klain, who had served as Vice President Joe Biden's chief of staff, according to people familiar with the administration's plans.