The United States, which has the largest share of votes in the IMF, the main institution overseeing global financial stability, and second biggest voter Japan are expected to officially endorse Lagarde as early as June 10 when the nomination process closes, Japan's Nikkei said.
Japanese officials were not available for comment. Finance Minister Yoshihiko Noda has only said a new IMF chief should be selected through an open, transparent and merit-based process.
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Emerging market countries argue that their increasing influence in the global economy, with China and India the world's fastest-growing nations, should give them a stab at the IMF top post, which has traditionally gone to a European.
Mexican central bank governor Agustin Carstens has also been nominated to succeed Dominique Strauss-Kahn, who resigned on May 18, but he lacks broad support.
The IMF has a June 30 deadline to pick a successor to Strauss-Kahn. The IMF board has said it hoped member countries would agree through consensus on a candidate although it could go to a vote.
The United States and Europe hold 48 percent of the votes in the IMF, and Japan has more than 6 percent. Assuming all European members unite behind Lagarde, she would have more than 51 percent of the votes and get the post, Nikkei said.
The top IMF job fell vacant after Strauss-Kahn was arrested and charged last month with the attempted rape of a maid in a New York hotel. He has denied the charges.
(Reporting by Tetsushi Kajimoto; Editing by Robert Birsel)