Unesco has long been viewed in the West as politicized, corrupt and anti-American, antipathy that came to a head in 1984 when President Ronald Reagan withdrew American membership. Since then, the agency has been much reformed, both in terms of its finances and its embrace of values like freedom of the press and education for women.

Hillary Rodham Clinton was the first American secretary of state to visit Unesco, coming this year to support an initiative on education for girls and young women, and Ms. Bokova emphasizes that since 9/11, Unesco has run its largest education project in Afghanistan, opening literacy centers for civilians as well as Afghan police officers. It cooperates on teacher training with American companies like Microsoft, she said, and has organized training for Tunisian and Egyptian journalists since the Arab Spring revolts.

Only this month, the United States made separate voluntary contributions to Unesco programs for education and clean water; Washington praises its work on behalf of universal literacy, gender equality and disaster preparedness.

If the United States withdrew its financing, it would still retain a seat at the agency for another two years, but even then its influence would be weakened.

“In a world where soft power is so important, the United States is counterproductively compromising its position in a forum that really matters,” said Ronald Koven, who monitors Unesco for the World Press Freedom Committee, an American nongovernmental organization.

Peter Yeo, vice president for public policy at the United Nations Foundation, which supports the organization’s goals, said that “what’s maddening is that this is not your grandfather’s Unesco — it is better managed, more efficient and U.S. leadership in Unesco has made it a better organization.”

Mrs. Clinton has asked the American special envoy to the Middle East, David M. Hale, to negotiate with the Palestinians and Arab countries to break the impasse. The State Department has said it hoped to press the Palestinians to withdraw their request.