Carstens is the governor of the central bank of Mexico, while Lagarde is the French finance minister, The New York Times reported Tuesday.
The IMF has traditionally chosen a leader from Europe, but emerging countries were vocal this year about having a leader chosen from a different region.
Out of the running is Stanley Fischer, governor of the Bank of Israel and a former IMF deputy director. Fischer said his candidacy had been scrubbed because the age limit for candidates is 65. Fischer is 67.
In a statement, Fischer said, "I think that the age restriction, which was set in the past at 65, is not relevant today."
It is possible Fischer was also ruled out because, as dual citizen of the United States and Israel, his directorship would have angered Arabs, the Times said.
Russian candidate Grigori Martchenko has also apparently been ruled out, which leaves Lagarde, a European, as the front runner.
Lagarde visited China, Brazil and India recently to build a consensus of support.
The new director will succeed Dominique Strauss-Kahn, who resigned in May after being arrested in New York City and charged with sexual assault of a hotel maid.
Strauss-Kahn not only lost his IMF directorship, but also his potential candidacy to become the next president of France.
Notable deaths of 2014 [PHOTOS]
Larry Ellison to step down as CEO of Oracle