In a meeting with Lagarde, who is on a two-day India visit, Modi "emphasized the need to have greater representation for Indians at the higher level of IMF, particularly at the Deputy MD level".
Earlier in the day, Lagarde had said during an event that "one of these days" an Indian can head the International Monetary Fund (IMF).
"I hope one of these days. I can't tell you when. One day of course," she said in reply to a query on when an Indian could head the IMF.
While Lagarde is from France, her predecessor Dominique Strauss Kahn at IMF was also from the same country. Traditionally, the position of IMF's MD has been held by a European, although there have been pressure in past few years from emerging countries from Asia and Latin America to change this trend.
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During her meeting with the Prime Minister later in the day, Lagarde "complimented the government on the focused way in which it has been working to revive the economy", according to a statement issued by the Prime Minister's Office.
"He said that his focus, while promoting growth, is to create jobs, have inclusive growth, and focus on development of backward regions in Eastern and North-Eastern India. This region is rich in natural and human resources and we need to capitalize on this. The priority is on all fronts - agriculture, manufacturing and services," the statement said.
Lagarde told the Prime Minister that that the results of the government's results were visible and India's growth rate has crossed 7.5 per cent.
Lagarde also called on President Pranab Mukherjee and met Finance Minister Arun Jaitley.