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From left, President Michelle Bachelet of Chile; Irina Bokova of Bulgaria, head of Unesco; Kristalina Georgieva of Bulgaria, a European Commission vice president; and Helen Clark of New Zealand, head of the United Nations Development Program, have been cited as possible nominees. as possible successors to Secretary General Ban Ki-moon.

UNITED NATIONS — The bearded diplomat in a striped tie addressed a roomful of his peers the other morning on the subject of how the next United Nations secretary general should be chosen.

Among the suggestions offered by the diplomat, Margus Kolga of Estonia, was to finalize the selection as early as three months before the new term begins, in January 2017, “simply in order for him to prepare,” as he put it.

“Him?” piped up Mary Robinson, a former president of Ireland, so quietly that Mr. Kolga did not hear her at first.

“Him?” she repeated, a bit louder.

Mr. Kolga threw up his hands in surrender and began to apologize.

“Him or her. Sorry,” he said, adding that in his native Estonian, “there are no genders. When I’m saying ‘he,’ I’m meaning he or she.”

Whether the next secretary general will be a he or a she has become an increasingly potent subject of conversation, both inside and outside the corridors of the United Nations. Three dozen countries, led by Colombia, are promoting the idea that it is a woman’s turn to lead the organization. Women’s groups have put out lists of candidates. Prominent world leaders — including members of the group Ms. Robinson belongs to, the Elders, composed of former heads of state — have called for countries to nominate women.

Women have been elected to lead countries as varied as Germany and South Korea. The International Monetary Fund is led by a woman, and some of Europe’s biggest companies are required by law to set aside 30 percent of their supervisory seats for women. Sweden’s foreign minister, Margot Wallstrom, has pursued “a feminist foreign policy,” calling women’s rights critical to global peace and security. And Hillary Rodham Clinton — who, 20 years ago, spoke as the first lady of the United States at a landmark United Nations women’s conference in Beijing — is running for president.

The United Nations, though, has been something of a holdout. Since its inception in 1945, it has always been led by a man.

“After eight male secretary generals in a row, the Elders are very sympathetic to the idea that it is high time for a woman to be chosen,” Ms. Robinson said when it was her turn to speak. “But if it turns out that the right candidate is a man, then so be it.”

The “right candidate,” she took pains to say, should be “independent and not beholden to the interests of individual member states.”

The calls reflect not only an appeal for gender equity but also a growing sense of frustration with the opaque way in which world powers — namely, the five permanent members of the United Nations Security Council: Britain, China, France, Russia and the United States — bargain over the choice of the world’s top civil servant. Diplomats and civil society activists say that if the process does not change, it stands to make the United Nations anachronistic, irrelevant and unfit to handle the most pressing global crises.

“Whatever the selection process for the next secretary general is, historically there’s been no attention paid to the representation of half the world’s population,” said Louise Arbour, a Canadian jurist who was the United Nations high commissioner for human rights from 2004 to 2008. “It is geography; then it is horse-trading on state interests, much more than the personal qualities of the candidate.”

Few countries have announced their nominations, and in keeping with the protocol of giving different regional blocs a chance, Eastern Europe is angling for its turn, though nothing in the United Nations Charter requires it.

Because of the calls for female nominees, many more of the names being talked about are of women than ever before. They include Irina Bokova of Bulgaria, director general of Unesco; President Michelle Bachelet of Chile; Kristalina Georgieva, another Bulgarian and a vice president of the European Commission; and Helen Clark, a former prime minister of New Zealand who heads the United Nations Development Program.

One of the men whose name has come up is Danilo Turk, a former president of Slovenia.

Ms. Clark declined to confirm in a telephone interview whether she was interested in the job, though some of her staff members described her travels to major capitals around the world as part of a campaign for the post. She said only that women were underrepresented in senior leadership positions in the United Nations, in sharp contrast to other organizations, and that women tended to bring different issues to the table when they were represented at the top.

“My perspective is that women bring a wider range of life experiences,” Ms. Clark said. “All top positions globally, women should have an equal chance to compete for them.”

The United Nations Charter does not specify exactly how to fill the top job. In practice, for most of its 70-year history, the Security Council has announced its preferred candidate, who has then been ratified by the broader membership of the General Assembly. The Council’s deliberations are private, and many diplomats complain that the Council’s permanent members try to make sure that their selection bends easily to their wishes — more secretary than general, the saying goes.

The current secretary general, Ban Ki-moon, is known as an advocate of promoting women to management posts, sometimes scolding heads of state and government for not putting more women in cabinet positions. Still, female peacekeepers remain scant, as do female mediators appointed by the United Nations to negotiate peace deals.

Among the ambassadors for the permanent members of the Council, only Britain’s, Matthew Rycroft, has explicitly said his government would favor a woman among equally qualified contenders. The Russian ambassador, Vitaly I. Churkin, has warned that men should not be discriminated against.

More than one-fourth of the ambassadors representing their countries at the United Nations are women. That is a far greater share than when Ms. Arbour first came to United Nations headquarters 18 years ago and watched the parade of diplomats and heads of state entering the General Assembly halls.

“Is there a separate entrance for women?” she recalled asking a veteran United Nations staff member. “No, honey, this is it,” she was told.

There is, however, a flip side to the chorus for a female secretary general.

“If you have a woman who doesn’t deliver, they will use her to judge the rest of us. That’s my worry,” said Zainab Bangura, the secretary general’s special representative on sexual violence in conflict. “What I’m looking for is an exceptionally good woman who can demonstrate that women can really do that job.”