Lagarde said she had "productive" one-on-one meetings with IMF executive directors Wednesday and early Thursday in Washington, before a three-hour interview with the 24-member board in the afternoon.
The 187-nation Fund needs to "continue its shift towards responsive, even-handed and balanced action in support of global economic and financial stability, the better to serve the whole membership," she said in a statement to the board.
"I am not here to represent the interest of any given region of the world, but rather the entire membership," she said.
Lagarde is the front-runner in the race for the IMF top left vacant after managing director Dominique Strauss-Kahn resigned May 18 to fight sexual assault charges in New York. Strauss-Kahn denies the allegations.
Lagarde's sole rival, Mexico's central bank governor Agustin Carstens, admitted that Lagarde's chances of being elected by the board "are quite high."
Lagarde has the backing of Europe, which holds seven of the 24 seats on the executive board, and the board hopes to select the new managing director by consensus.
Under a tacit agreement between the United States and Europe, the leadership of the IMF has always gone to a European, while the top job at the World Bank has always gone to an American.
The United States, the largest IMF stakeholder with nearly 17 percent of the vote, has not publicly endorsed either Lagarde or Carstens, considered an emerging-market candidate.
Lagarde met with US Treasury Secretary Timothy Geithner Thursday morning before heading to the board interview.
Geithner "believes that Minister Lagarde's strong leadership skills and experience makes her an exceptionally talented candidate for IMF managing director," Treasury spokeswoman Natalie Wyeth said.
"He again reiterated his appreciation for the strong candidates who have come forward, particularly at this critical time for the global economy," she said.
Lagarde emerged from IMF headquarters after the board audition looking relaxed.
"I believe the Fund should be more responsive, certainly more effective and more legitimate," she told a gaggle of waiting reporters.
"And that really entails many potential improvements but also continuation of the reforms that have been undertaken by my predecessor," she said, her choice of word for fellow countryman Strauss-Kahn suggesting she feels confident about her selection as the new boss.
Lagarde said the IMF was a "remarkable" institution with an "exceptional staff."
"It should become more responsive, more involved, more legitimate with regard to the integration of its 187 members. In any case, that would be my goal," she added. "It's now for the membership to decide."
The IMF board has set a tight deadline for filling the top job: June 30.
The executive board meets Tuesday with the aim of reaching a decision "by consensus," according to the IMF, but in case of deadlock it will decide by majority vote.
The announcement of the new managing director could come before next Thursday's deadline, IMF spokesman David Hawley said.
Lagarde's interview with the board followed on the heels of Carstens's two-day campaign at IMF headquarters. He also had a three-hour board interview Tuesday and the IMF issued his statement to the board.
Lagarde, 55, would be the first woman to head the Washington-based IMF, which oversees the global financial system and provides emergency lending to medium- and high-income member countries.
A lawyer by training, she would also be the first IMF managing director who was not an economist.
Her rival Carstens, 53, is an economist who has worked at the IMF, including a stint in the number-three position, deputy managing director, from 2003 to 2006.
Lagarde tackled head-on criticism that she, as a European, would have a conflict of interest in dealing with IMF involvement in the financial rescues of Greece, Ireland and Portugal.
"I am on record for having supported a selection process regardless of nationality. As a consequence, being French and being European should be neither an advantage nor a handicap," she said in her statement to the board.
The IMF said no start date has been set for the next managing director.