With her term coming to an end in July, the International Monetary Fund formally began accepting nominations for who will guide the global crisis lender for the next five years.
Lagarde, who has led the IMF through one of its most difficult challenges, the rescue of the eurozone from meltdown, has only said that she is open to another term.
But she drew immediate endorsements from Germany, France, Britain, and the Netherlands, while the United States praised her performance in the role without giving its formal backing.
He described her as "an outstanding leader with vision and acumen to steer global economy in years ahead".
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German Finance Minister Wolfgang Schaeuble also backed Lagarde's candidacy.
She had proven to be a "far-sighted and successful crisis manager in difficult times," the German finance ministry in Berlin said in a statement.
The Netherlands's finance minister, Jeroen Dijsselbloem, tweeted that his country "fully supports her candidacy," while France, where Lagarde was minister of finance before taking the IMF job, also endorsed her.
But he expressed strong approval of her performance.
"I have a very close working relationship with her, (and) the highest regard for her," Lew said. "I think she has done a great job, you know. I look forward to continuing to work with her."
On the first day that nominations for the powerful position were open, there were no other names being mentioned as candidates. But the Fund will be taking nominations through February 10. After a review of the candidates, the IMF executive board aims to have decided on a candidate by March 3.
But her win came amid criticisms that the IMF's top job should not be locked down by a European, as it has since the institution was created in 1944.