WASHINGTON — When Russia and Ukraine reached an agreement on Friday to unblock Ukrainian grain exports, Turkey’s president, Recep Tayyip Erdogan, played the role of benevolent statesman.

Seated next to the United Nations secretary general in an Ottoman palace in Istanbul, Mr. Erdogan said the deal, which Turkey helped to broker, would benefit “the whole of humanity.”

President Biden’s administration welcomed the agreement, which could relieve a global food crisis intensified by Russia’s invasion of Ukraine and blockade of its ports. Officials expressed skepticism about whether Russia was acting in good faith, and Russian missiles struck the Ukrainian port city of Odesa less than a day after the pact was signed. Still, a White House spokesman had commended Mr. Erdogan for his efforts.

But privately, Mr. Erdogan has remained a source of substantial irritation for Biden administration officials.

Days before presiding over the grain agreement, the Turkish autocrat renewed a warning that he might veto NATO’s plans to accept Sweden and Finland as members in the coming months, an act that would deeply embarrass the alliance and the Biden administration as they work to counter Russia. And Congress this month expressed misgivings about Mr. Biden’s pledge at a NATO summit in Spain last month to sell dozens of F-16 fighter jets to Turkey.

On Tuesday, Mr. Erdogan traveled to Tehran for meetings with Iran’s president, Ebrahim Raisi, and Russia’s president, Vladimir V. Putin. The images of two prime American rivals with Mr. Erdogan, the leader of a NATO country, clashed with the Western narrative of a deeply isolated Iran and Russia, analysts said.

Then on Friday, a White House spokesman reiterated U.S. concerns about Mr. Erdogan’s threats to mount a new invasion of northern Syria targeting U.S.-backed Kurdish fighters whom he considers terrorists.

Taken together, Mr. Erdogan’s actions — and Mr. Biden’s limited ability to restrain them — underscore the Turkish leader’s unique position as a military ally frequently at odds with the agenda of his Western allies. To U.S. officials, it is an often maddening role.

“Erdogan is basically the Joe Manchin of NATO,” said Elizabeth Shackelford, a former Foreign Service officer, referring to the conservative Democratic senator from West Virginia who has stymied Mr. Biden’s domestic agenda. “He’s on our team, but then he does things that are so clearly not good for our team. And I just don’t see that changing.”

But Biden administration officials say that writing off Mr. Erdogan entirely would be self-defeating. His nation’s position at the crossroads of East and West is strategically important and allows him to be an interlocutor with even more troublesome neighbors — as evidenced by the grain deal, which created a demilitarized corridor through the Black Sea for Ukraine’s agricultural exports.

A senior U.S. official said that much of Mr. Erdogan’s problematic behavior was a function of his political weakness in Turkey, where the inflation rate climbed to almost 80 percent last month. Hoping to shift attention from his mismanaged economy, Mr. Erdogan has turned to chest-thumping displays of nationalism and demagogy over the threat from the P.K.K., a Kurdish separatist movement in Turkey, and Kurdish groups in Syria.

Major NATO initiatives, like the proposed expansion of the 30-member alliance to include Sweden and Finland, require unanimous consent. Mr. Biden said in May that he hoped the two countries could “quickly” join in what would be a major strategic blow to Mr. Putin.

But Mr. Erdogan raised objections, complaining that both potential new members have lent political and financial support to the P.K.K., which the United States has designated as a terrorist organization because of its history of violent attacks. U.S. and NATO officials worried that the planned expansion could collapse in a major propaganda win for Mr. Putin, who has long worked to divide the alliance.

NATO leaders heaved sighs of relief at their summit last month when Mr. Erdogan reached an agreement with the leaders of Sweden and Finland, who pledged to act against terrorist organizations and join extradition agreements with Turkey, which wants to prosecute P.K.K. members living in those countries.

Mr. Biden seemed especially grateful for the breakthrough. “I want to particularly thank you for what you did putting together the situation with regard to Finland and Sweden,” he told Mr. Erdogan in the presence of reporters.

The two-page agreement said in generalized language that Sweden and Finland would address Turkey’s “pending deportation or extradition requests of terror suspects expeditiously and thoroughly.” But Turkish officials have said they expect the extradition of more than 70 individuals. It was unclear whether Sweden and Finland would agree or how Mr. Erdogan might react if they did not.

On Monday, Mr. Erdogan warned that he could still “freeze” NATO’s expansion if his demands were not met.

Mr. Biden also told Mr. Erdogan in Spain that he supported the sale of 40 American F-16 fighter jets that Turkey requested last fall, along with technology upgrades for dozens of fighters it already owns. Turkey wants those planes in part because the Trump administration canceled plans to sell the country advanced F-35 fighter jets in 2019 after Mr. Erdogan, in one of his more confounding recent moves, purchased Russia’s S-400 antiaircraft missile system in defiance of U.S. warnings.

Mr. Biden denied that he offered the planes to buy Mr. Erdogan’s support for NATO’s expansion. “And there was no quid pro quo with that; it was just that we should sell,” he said. “But I need congressional approval to be able to do that, and I think we can get that.”

Congress’s approval may not be a given. And it was unclear whether Mr. Erdogan might block NATO’s proposed expansion until he reaches a deal on the F-16 jets.

This month, the House approved an amendment to an annual military policy bill requiring Mr. Biden to certify that any sale of the fighter jets is in America’s vital national interests and that Turkey will not use the jets to violate the airspace of Greece, its Aegean Sea neighbor and fellow NATO ally, with whom Ankara is engaged in a bitter territorial dispute.

Representative Chris Pappas, Democrat of New Hampshire and the amendment’s sponsor, also cited Mr. Erdogan’s purchase of the Russian missile system and equivocal position toward Russia’s invasion of Ukraine. Mr. Erdogan has called the invasion “unacceptable” but has not joined sanctions imposed by the United States and its allies on Russia.

“Enough is enough,” Mr. Pappas said. “Turkey has played both sides of the fence in Ukraine. They have not been the reliable ally that we should be able to count on.”

“I think the Biden administration needs to take a stronger stance,” he added.

Once the White House formally requests that Congress approve the sale of the planes, Mr. Biden will need the support of other influential members who have been highly critical of Mr. Erdogan, including the Democratic chairman of the Senate Foreign Relations Committee, Bob Menendez of New Jersey.

Mr. Menendez has previously questioned whether Turkey belongs in NATO at all. And at a hearing last month on the proposed NATO expansion, he said that “with time of the essence, the 11th-hour concerns by Turkey standing in the way of this process only serve Putin’s interests.”

Mr. Menendez also issued a statement last month with his Republican counterpart on the Foreign Relations Committee, Senator Jim Risch of Idaho, sternly warning Mr. Erdogan against his threatened invasion of northern Syria. They were joined by the Democratic chairman of the House Foreign Affairs Committee, Representative Gregory W. Meeks of New York, and his Republican counterpart, Representative Michael McCaul of Texas.

In the statement, the lawmakers said the potential invasion would have “disastrous results,” threatening local operations against the remnants of the Islamic State and exacerbating Syria’s humanitarian crisis.

A Pentagon official recently added to the American warnings.

“We strongly oppose any Turkish operation into northern Syria and have made clear our objections to Turkey,” Dana Stroul, a deputy assistant secretary of defense, said this month at the Washington Institute for Near East Policy. “ISIS is going to take advantage of that campaign.”

Some of Mr. Erdogan’s harshest critics warn of an endless cycle, in which the Turkish leader wins concessions from the United States and other NATO allies, such as new fighter jets and a tougher line against Kurdish militia fighters, only to escalate his demands in the future.

“This dance around the F-16 — it’s jet fighter diplomacy, and that is a mask of what’s truly at play here,” said Mark Wallace, founder of the Turkish Democracy Project, a group highly critical of Mr. Erdogan and his turn to authoritarianism. “A good ally — much less a good NATO ally — doesn’t use blackmail to get what it wants at key moments in the alliance’s history.”

Julian E. Barnes contributed reporting from Aspen, Colo.