Officially, the 24 members of the executive board of the IMF choose from a list of candidates for the post, with votes weighted according to the subscriptions of member countries and regions.
Other potential candidates:
- Gordon Brown, 60, former British prime minister;
- Axel Weber, 54, former German central bank chief;
- Tharman Shanmugaratnam, 54, Singapore Finance Minister and Deputy Prime Minister;
- Montek Singh Ahluwalia, 67, Indian planner, former World Bank employee and
director of the IMF's Independent Evaluation Office;
- John Lipsky, 64, American acting IMF managing director;
- Agustin Carstens, 52, Mexicand central bank governor and former finance
- Trevor Manuel, 55, South African former finance minister;
- Stanley Fischer, 67, Israeli ex-World Bank economist, now the head of Israel's central bank.
In practice, until now, there has been a pact between the US and Europe that an American should always hold the presidency of the World Bank, while a European gets the job of IMF Director General.
But the balance of power and economic clout in the world is shifting and emerging economies are pushing for an end to that cosy arrangement.
In November, G20 leaders agreed to reform the make up of the IMF board, and on Thursday Japan's finance minister said that the new IMF boss should be chosen in an "open, transparent" way.
However, Europe, as the biggest contributor to the IMF, will fight hard to make sure the post still goes to a European.
France's current finance minister, Christine Lagarde, appears to be emerging as a top contender.
The 52-year-old former lawyer who worked for many years in a leading United States Law firm, was tipped as a possible successor to Strauss-Kahn even before his current troubles.
She is thought to have the support of German Chancellor, Angela Merkel, and a highly placed European official said this week that London considered Lagarde certainly a credible candidate.
She declared on Thursday that Europe should unite around a single candidate, but did not say whether or not she herself would be interested.
Lagarde is facing a judicial investigation over whether she abused her ministerial position in playing a role in a case which resulted in a substantial payment to controversial French businessman Bernard Tapie.
But over the coming days support could coalesce around several other possible candidates.