Mr. Obama and his chief of staff, William M. Daley, have been urging Mr. Geithner to stay, administration officials say, not only for continuity when the economy has weakened and to avoid an all-but-certain confirmation fight in the Senate over a successor, but also because Mr. Obama has developed a close rapport with Mr. Geithner.

Whether the president persuades Mr. Geithner to stay will be a central development for the White House as it girds for a re-election race expected to turn on the economy and the continuing battle of the budget with Republicans…

Especially in recent weeks, the issue has become a running joke, officials say: Mr. Geithner and Mr. Daley tease about the ankle bracelet that the White House makes him wear, or Mr. Geithner asks if Mr. Daley has yet read his resignation letter, to which Mr. Daley answers in unprintable language.