"The ability of Strauss-Kahn to navigate those waters will not be easily replaced," said Uri Dadush, a senior associate at the Carnegie Endowment for International Peace and former director of economic policy at the World Bank.

Strauss-Kahn hasn't yet officially stepped down. But few analysts expect him to remain.

Morris Goldstein, a top economist at the IMF for 25 years and now at the Peterson Institute, said the French will likely argue that Lagarde would be a wise choice because the IMF will need someone closely involved in negotiations over aid packages for Greece, Ireland and Portugal. If chosen, Lagarde would be the first woman to lead the IMF.

"They could argue that at this delicate state of negotiations on the European debt programs, the IMF needs somebody who is familiar with the issues," Goldstein said.

He said that if the Europeans lose the IMF post, they might insist that the United States also give up its long-standing hold on the World Bank job when Robert Zoellick's term as World Bank president ends June 30, 2012.

Non-Europeans will likely argue that it's time to open up the leadership to major developing countries whose economies are expanding far faster than the developed world. Some, like China, have boosted their contributions to the IMF in recent years.

"There is growing disquiet, particularly among emerging nations," said Jan Randolph, an analyst at IHS Global Insight. "China could use its influence to support an emerging-market candidate for the top IMF job."

The United States, the largest donor to the IMF, is certain to weigh in.

"This will be a rather complicated game of musical chairs," said Uri Dadush, a senior associate at the Carnegie Endowment for International Peace and former head of economic policy at the World Bank.