By Lesley Wroughton
While Carstens will get the backing of Latin America, Canada and Australia, his support appears too thin to break Europe's 64-year hold on the IMF post.
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The 24-strong IMF board meets on Tuesday to finalize a process that began in May after Dominique Strauss-Kahn resigned as IMF managing director to defend himself against charges of sexual assault and attempted rape. He denies the charges.
IMF board directors, who represent the fund's 187 member countries, want to try to reach a consensus decision on a successor that would allow them to avoid a formal vote. It is possible the process could spill into Wednesday.
The race has been one of the most hotly contested succession battles in IMF history.
In a convention dating back to the creation of the IMF and World Bank after World War Two, Europe has always held the top IMF job, while the World Bank's top post has always gone to an American.
Developing countries have warned against another U.S.-European stitch-up, but some potential candidates from emerging markets decided not to step up because they did not feel they had a fair chance at the job.
Although a long-shot candidate, Carstens vigorously campaigned on his experience as a former IMF official who had first-hand knowledge of developing world economic crises.
An important voice in Tuesday's discussion will be the United States, which has been silent on who it supports.
U.S. Treasury Secretary Timothy Geithner said the selection proceed had produced "two very credible candidates" but he has also stressed the need for a quick resolution to the decision.
However, the Obama administration is widely expected to back Lagarde to preserve America's hold on both the top World Bank job and the No. 2 spot at the IMF.
Washington holds close to 17 percent of the vote, while European nations account for around 40 percent. China and Russia, two of the so-called BRICS emerging market countries, have already said they back Lagarde.
The three other BRIC nations -- India, South Africa and Brazil -- have been quiet on who they support.
Countries such as Egypt, Saudi Arabia, Indonesia, South Korea and French-speaking African nations early on declared their support for Lagarde, but they have only marginal voting power.
Fears of contagion over an escalating debt crisis in Greece have played in Lagarde's favor over the last several weeks because of her political punch across Europe, IMF board officials said.
A few board directors quietly expressed concern over an unresolved legal investigation into Lagarde's role in a 2008 arbitration payout to a French business. A top French court has put off a decision on the matter until July 8.
One way of dealing with the issue is not to offer Lagarde an IMF contract until the court has made a final decision, a board source suggested.
Whichever candidate steps in will have to immediately be able to deal with further IMF-EU financing to keep Greece afloat and focus on potentially thorny IMF spillover reports that analyze the economic and policy actions among the world's major economies.
Lingering resentment over the outcome of the latest election process will also require the new managing director to act quickly to reassure developing nations they have a stake in decision-making at the Fund.
(Editing by Paul Simao)