“She has been a very effective leader,” said Edwin M. Truman, a specialist in international finance formerly at the Federal Reserve and the United States Treasury. “Yes, there are big questions about the fund’s future. But for her to have to step down now — well, that would be complicated.”

Jacob J. Lew, the Treasury secretary, expressed the Obama administration’s support for Ms. Lagarde, saying that “she is a strong leader of the I.M.F., and we have every confidence in her ability to guide the fund at a critical time for the global economy.”

For the Trump administration, “I don’t think this kind of ethical question is likely to be the highest priority,” Mr. Truman said. While the I.M.F. and other global institutions did not figure in the presidential debate, Mr. Trump repeatedly criticized a “global power structure” that fixed the economy against workers.

“At bottom, it’s all about French politics,” Mr. Truman said.

Members of the I.M.F. board were well aware that Ms. Lagarde was facing trial in her native France over allegations that occurred when she was the finance minister in the administration of Nicolas Sarkozy.

The consensus among the directors was that Ms. Lagarde’s transgressions occurred when she was not at the fund — in contrast to those of her predecessor, Dominique Strauss-Kahn — and since taking charge in 2011, she had proved to be a leader capable of presenting a softer side of the fund while fighting hard to bolster its legitimacy in the aftermath of the financial crisis.

More so than her predecessors, Ms. Lagarde has pushed the fund to be more aggressive in taking up the cause of women and focusing attention on growing issues of inequality around the world.

Over the last year and a half, she has also led a forceful public critique of Europe’s refusal to offer Greece debt relief in return for the difficult economic changes the country has been making.