Geithner, who turns 50 Aug. 18, has not yet notified the White House of his intentions, and family considerations could still win out, advisers tell The New York Times.
Geithner -- the last remaining member of Obama's original team of economic advisers that shaped the president's response to the financial crisis -- said in early July he would consider resigning after a deal was reached to increase the nation's debt ceiling and reduce the federal deficit. Obama signed that deal into law Tuesday.
Obama and Chief of Staff William Daley have been urging Geithner to stay, in part because Obama has developed a close rapport with him, administration officials told the Times.
Obama and Daley also want continuity during the current tough economic times and to avoid an all-but-certain confirmation fight in the Senate over a successor, the newspaper said.
The former co-chairman of Obama's debt panel, Erskine Bowles, who was White House chief of staff under President Bill Clinton, is one of several candidates who were being considered if Geithner stepped down, officials said.
Other candidates included investment banker and former deputy treasury secretary under Clinton, Roger Altman, Federal Reserve Board Vice Chairwoman Janet Yellen and National Economic Council Director Gene Sperling, who previously served as a Geithner lieutenant, the officials said.
Geithner told ABC News Tuesday he hadn't decided whether he would leave.
"We've got a lot of challenges, the president's got a lot of challenges and, you know, I got other pressures on me too," he said.
Geithner's wife and son moved back to New York in June so their son could finish high school where he grew up.
Before moving to the Washington area, the Geithners lived in Larchmont, N.Y., north of New York City, when Geithner was president of the Federal Reserve Bank of New York.
Geithner said privately last month he would not ask to leave without Obama's agreement.