Poorer nations that contribute to the I.M.F.’s financing have grumbled about having to prop up rich Europe. More than half of the I.M.F.’s lending goes to the euro zone, from virtually nothing a few years ago. The I.M.F. has contributed about a third of the money used to rescue countries like Portugal, Ireland and Greece, with the rest coming from other euro zone countries.

“Historically, Europe took no I.M.F. lending,” said Guntram B. Wolff, the deputy director of Bruegel, a research organization in Brussels. “Now lending has increased since the beginning of the crisis dramatically. Is it appropriate? That is a very big question.”

Leaders and citizens of countries like Greece, Portugal and Ireland have complained bitterly about the terms that the I.M.F., as part of the so-called troika of technocrats along with the European Central Bank and the European Commission, has imposed in return for loans.

In addition to budget cuts and tax increases, governments have been pressured to roll back rules that protect some workers from dismissal and impose other unpopular changes. Even if the I.M.F. is rethinking its stance on austerity, it will continue to demand strict conditions because that is the only leverage the organization has to get its money back.

Ms. Lagarde, the former finance minister of France, is perceived as less doctrinaire than the Germans, but she was at the table last month when leaders negotiated an ill-fated plan to make ordinary bank depositors help pay for a bailout in Cyprus. Although the I.M.F. had reservations about imposing a levy on insured depositors in Cyprus, Ms. Lagarde went along with the accord. After an outcry, the plan was revised to put the burden on large depositors.

But even those who have doubts about the I.M.F.’s role in Europe see no alternative. The organization will inevitably be a force in Europe for years to come, because of the money that it has lent and because of its traditional role as watchdog over the economic and budget policies of its members.

“If the I.M.F. wasn’t participating at all, the crisis would have been worse,” said Morris Goldstein, a former deputy director of research at the I.M.F. who is now a senior fellow at the Peterson Institute for International Economics, a research organization in Washington.