Head of the International Monetary Fund, France's Christine Lagarde, arrives at the French Republic Justice Court, on May 23, 2013 in Paris, for a hearing about her role in the settlement of a multi-million-dollar business dispute when she was French finance minister in 2008. (Lionel Bonaventure/AFP/GETTY IMAGES)

Christine Lagarde, managing director of the International Monetary Fund, submitted to more than 12 hours of interrogation Thursday about her role in the settlement of a multimillion-dollar business dispute when she was France’s finance minister in 2008.

Lagarde, smiling and elegant in a dark suit and turquoise scarf despite a long day, was ordered to return Friday for more grilling by a trio of investigating magistrates at the Court of Justice of the Republic, a judicial panel that handles malfeasance cases in France.

“Until tomorrow,” she told cameramen as she got into a waiting car and was driven away.

The magistrates were querying Lagarde, 57, about her actions during a $520 million settlement granted by an arbitration board to the maverick French businessman Bernard Tapie. The payout ended a long and involved legal dispute between Tapie and the Credit Lyonnais — which previously was in the courts — as the bank was being liquidated after failing and being taken over by a state governing board.

At the end of the interrogation, Lagarde could be retained as a witness as the investigation proceeds or, if the magistrates deem they have credible evidence warranting such a step, they could cite her as a suspect in malfeasance. This would raise the question of whether Lagarde could remain as head of the IMF, the Washington-based financial institution that plays a major role in the world economy, including anticorruption efforts in financially ailing countries.

Najat Vallaud-Belkacem, the French government spokeswoman, said on the all-news BFM-TV station that she supposed Lagarde would have to step down if she were being investigated as a suspect. The IMF board has expressed confidence in her, however, and President Francois Hollande reportedly told her two months ago that he would back her through the investigation.

“The executive board has been briefed on this matter, including recently, and continues to express its confidence in the managing director’s ability to effectively carry out her duties,” IMF spokesman Gerry Rice said.

Lagarde took over at the IMF in 2011 after the disgrace of Dominique Strauss-Kahn, also a former French finance minister, who was involved in several sex scandals. As a result, several political figures have expressed fears of another blotch on France’s image.

Lagarde, a former international lawyer based in Chicago, repeatedly has said that she made the decision to go to arbitration on her own and that it was in the best interests of the government. The underlying question, however, is whether she acted at the behest of former President Nicolas Sarkozy, who appointed her, or of his chief of staff, Claude Gueant, in what would have amounted to political favoritism to Tapie.

Howard Schneider in Washington contributed to this report.