UNITED NATIONS — He’s the new leader of the United Nations, an international diplomat who spent years focused on the plight of the world’s refugees. She’s a diplomatic neophyte representing an “America First” administration that seeks travel bans for refugees and mocked the United Nations.

It is an awkward relationship. But as the White House stands poised to slash funding for the United Nations — how severely is still to be determined — it is a critical relationship for both the secretary general, António Guterres, and the United States ambassador, Nikki R. Haley.

For Ms. Haley, an ambitious politician, the ambassadorship is an opportunity to burnish her foreign policy credentials. Mr. Guterres, a former politician himself, needs to fend off the decimation of his organization.

For the last six weeks, the two diplomats have been dancing around each other delicately, she in her trademark pointy-heeled boots, he in avuncular brogues, both trying to leverage the other to get what they need.

Ms. Haley, 45, a former governor of South Carolina, has a lot to gain. She is expected to have a long political career ahead of her, and being the envoy to the United Nations allows her to build on her limited foreign affairs experience.

As the United States ambassador, she has enormous leverage over the United Nations, but she also appears to have some freedom to speak her mind. She has criticized Russia’s role in the Syrian conflict, using language that differed markedly from that of President Trump, who has adopted a more conciliatory stance toward Moscow.

She has also echoed the Trump administration’s criticism of the United Nations, deploring what she said was its anti-Israel bias and its lack of efficiency, even as she has taken pains to build good will among her diplomatic peers.

In a statement on Thursday, she said she was “working closely” with Mr. Guterres to reform the United Nations and “restore trust and value.” She signaled clearly that the organization would face funding cuts, saying, without providing details, that “in many areas, the U.N. spends more money than it should.”

For Mr. Guterres and the United Nations, the stakes are extremely high, as they are for people around the world who depend on the organization for such diverse things as childhood vaccinations, food in times of famine, peacekeepers to protect them from marauding armies, and envoys who try to bring warlords to the negotiating table. (The stakes are less critical for the United States, at least fiscally speaking: Funding for the United Nations is less than 0.1 percent of the total federal budget.)

And so, Mr. Guterres, who took office Jan. 1, has pulled out all the stops to engage Ms. Haley.

He hosted her for lunch in his 38th-floor dining room on the day she presented her credentials and promised a muscular American approach to diplomacy. “We’re taking names,” she said about countries that crossed the United States, before going upstairs to meet him. Mr. Guterres has since met with her at least a half dozen times and spoken to her by phone on other occasions.

A former prime minister of Portugal, Mr. Guterres, 67, has tried to cast himself as the man who can deliver a leaner, nimbler United Nations — and therefore one deserving of United States support. He has deferred to the United States — perhaps too much, his critics have said. And while he has criticized the global tide of populism in generic terms, he has said little directly about Mr. Trump’s pronouncements or his policies.

“I think the S.G. has found the right balance,” the French ambassador, François Delattre said on Thursday, using United Nations parlance for the secretary general. “I believe he has established excellent relations with Nikki Haley.” Mr. Delattre added: “I believe that for the S.G., and for us, too, the U.N. reform is a key priority and will remain so, which helps in current circumstances.”

Ms. Haley has brought a notable personal style to the job.

She has rebranded the United States Mission as #TeamHaley on Twitter. She has gushed about how much she loves “The Americans,” the spy drama about Russian espionage. She has drawn criticism from rights groups for inviting the Center for Family & Human Rights, a group that opposes gay rights and is listed as a “hate group” by the Southern Poverty Law Center, to be part of the United States delegation to an annual women’s rights meeting at the United Nations.

She has criticized the United Nations Human Rights Council for welcoming countries like Cuba and China, though she has made no moves to withdraw the United States from the council.

Repeatedly, she has homed in on a purely domestic audience. She said in her confirmation hearing that she did not favor a “slash and burn” approach to cutting funding for the United Nations, but she then questioned whether the United States gets what it pays for.

The United States is by far the largest donor to the United Nations in a number of ways.

First, it pays dues, like every other country, based on its wealth, which means that the United States pays 22 percent of the organization’s operating budget. Second, because it is a veto-wielding member of the Security Council, it pays a higher share of the peacekeeping budget — 28 percent currently, which Mr. Trump wants to reduce to 25 percent. And third, it pays voluntarily for perennially underfunded aid agencies like the World Food Program.

Steep cuts, warned diplomats and humanitarian aid groups, would devastate not only the United Nations, but also the United States’ standing in the world — along with Ms. Haley’s.

On Instagram, A Better World Campaign, an advocate of United States funding for the United Nations, posted a graphic showing that the world body provides food to 80 million people. Ms. Haley clicked “like.”

Ms. Haley has said little about where she favors cuts, except that she will go through each of the peacekeeping missions with a fine-tooth comb.

That poses a test for Mr. Guterres, most immediately at the end of the month, when the mandate for his largest and most expensive operation, the $1.2 billion peacekeeping mission in the Democratic Republic of Congo, comes up for renewal. It remains to be seen whether Ms. Haley will push him to sharply reduce its size, or worse, shut it down entirely.

Ms. Haley’s toughest remarks have been on the Israeli-Palestinian conflict.

She has excoriated the Security Council for discussing the conflict on a monthly basis and forced Mr. Guterres to withdraw the nomination of a United Nations envoy who is a Palestinian. It was seen as an unprecedented act of deference to the United States by a secretary general.

And she has repeatedly criticized the Security Council for a resolution, adopted in December, that condemns Israeli settlement building and asks the secretary general to provide quarterly briefings on the subject.

In an effort to placate Washington, Mr. Guterres has opted not to present a written report at all. He is leaving it to his Middle East envoy, Nickolay Mladenov, to make an oral presentation — playing down the significance of a resolution that the Americans deplore.

Suzanne Nossel, a former senior official at the United States mission who is currently the executive director of the Pen American Center, said the challenge for Mr. Guterres now is to show the world, and principally the Americans, what kind of secretary general he will be: one who tries to keep powerful countries happy, like his predecessor, Ban Ki-moon, or one whose tussles with the powerful cost him his job, like Boutros Boutros-Ghali in the 1990s. Will he be a “vessel” for member states, she asked, or a moral counterweight?

“If he takes that shot now and takes on as antagonists all the populists and authoritarians around the world,” Ms. Nossel said, “it could be a long five years.”