Mr. Trump’s gamble was based on his calculation that Mr. Xi, the Chinese president, could put heavy pressure on North Korea to curb its nuclear weapons and missile programs. To secure Mr. Xi’s cooperation, the president soft-pedaled his harsh stance on China’s trade practices, and has said little about its adventurism in the South China Sea.
But a growing number of Mr. Trump’s aides fear that the bet is not paying off.
China has not significantly tightened the pressure on North Korea since Mr. Trump met with Mr. Xi in Palm Beach, Fla., in April. Its failure to do more has frustrated White House officials, who plan to raise the issue with their Chinese counterparts at a high-level meeting here on June 21.
China’s reluctance to exert its influence in this regard has left the Trump administration with few good options in dealing with the North Korea crisis. And it may lead the administration to try to negotiate with North Korea’s leaders, an approach that did not work for Mr. Trump’s predecessors but which he has periodically seemed to embrace.
The White House stepped gingerly in the direction of engagement this week when it sent a senior diplomat to North Korea to obtain the release of Otto F. Warmbier, a gravely ill American prisoner. The encounter, some former American officials said, could lay the predicate for further such conversations.
But North Korea’s inhumane treatment of Mr. Warmbier, 22, complicates the prospects for diplomacy. At an emotional news conference on Thursday, Mr. Warmbier’s father said his son had been “brutalized and terrorized” during 18 months of captivity; doctors said the son had sustained a severe neurological injury.
Still, current and former officials said the administration was unlikely to shift its strategy because of the mistreatment of a single American, no matter how horrific. The most likely outcome, they said, is that any future contacts with North Korea will be conducted in the utmost secrecy.
“The Trump administration has made it quite clear that this is an all-of-the-above approach,” said Daniel R. Russel, who served as assistant secretary of state for East Asian affairs under President Barack Obama. “They will explore what kind of official or unofficial contacts with North Korea can advance the cause.”
Mr. Trump’s aides are also motivated by South Korea’s recent election, which brought to power a progressive leader more interested in engagement than confrontation with the North. The president, Moon Jae-in, halted the deployment of an American antimissile system.
The Chinese are among those most interested in a meeting between Mr. Trump and the North Korean leader, Kim Jong-un, officials said, in part because that would put the onus for a solution on Washington, not Beijing. For the moment, though, the White House is more focused on getting China to put pressure on its neighbor.
Mr. Trump came into office promising to challenge China on a range of issues. But the imminent danger posed by North Korea’s nuclear and missile programs changed that calculus.
In April, he said Mr. Xi would be his partner in curbing Mr. Kim. Two weeks later, the Treasury Department declined to designate China as a currency manipulator, something Mr. Trump had promised to do during his presidential campaign.
Since then, there is scant evidence of a crackdown by Beijing on Chinese trade with North Korea. The Chinese government has not announced the shutdown of any major trading companies, nor have traders in Dandong, the Chinese port city on the North Korean border, reported any signs of movement against their companies. The North Koreans have fired 16 missiles this year, nine of them since the summit meeting between Mr. Trump and Mr. Xi.
“It is worrisome that the president seems to think he has convinced Xi Jinping to change course,” said Michael J. Green, a senior adviser on Asia in the National Security Council of President George W. Bush. “That’s the position he has staked out publicly, and it has huge implications for the region. He should not have oversold it.”
Mr. Trump’s quid pro quo has sowed concern in Japan and Southeast Asia, where officials are worried that the United States is muting its response to China’s aggressions in the South China Sea or could put off arms sales to Taiwan — actions that administration officials deny.
To raise the pressure on China, administration officials said the United States was considering sanctions on a range of Chinese individuals and banks that do business with North Korea — a step that could aggravate tensions with China and sour Mr. Trump’s fledgling relationship with Mr. Xi.
The next opportunity for the two leaders to meet face to face is at the Group of 20 meeting in Hamburg, Germany, in July. But a senior official said China’s responsiveness on North Korea could affect whether the White House schedules a one-on-one meeting.
It is not clear that even these steps would prod China to radically alter its approach to North Korea, which is driven less by concern over its nuclear ambitions than a dread of what a collapsed North would mean for China, with which it shares an 880-mile-long border.
Impatience with China has simmered for weeks inside the White House. But only in recent days has it begun to show.
“We cannot allow China to use its economic power to buy its way out of other problems, whether it’s militarizing islands in the South China Sea or failure to put appropriate pressure on North Korea,” Secretary of State Rex W. Tillerson said last week in Sydney, Australia.
Mr. Trump himself has withheld public judgment of Mr. Xi. At a rally in April to mark his first 100 days in office, the president said it would have been counterproductive to designate China as a currency manipulator at a time when he was seeking the government’s help on North Korea. And he said he had developed a sense for Mr. Xi over hours of talks.
“He’s a good man,” Mr. Trump told his supporters. “Now, he’s representing China. He’s not representing us. But he’s a good man. And I believe he wants to get that situation taken care of. They have tremendous power, and we’ll see what happens.”
In his Twitter feed, the president has veered between optimism and mild finger wagging toward China.
On April 21, he wrote, “China is very much the economic lifeline to North Korea so, while nothing is easy, if they want to solve the North Korean problem, they will.” On May 29, after another missile test, Mr. Trump wrote: “North Korea has shown great disrespect for their neighbor, China, by shooting off yet another missile … but China is trying hard!”
In the last month, traders said Chinese exports to North Korea were being inspected more thoroughly. The provincial authorities have warned traders not to hide banned goods that can be used in the North’s nuclear program — chemicals, for example — in general cargo.
Some longtime diplomats predicted that the Chinese would announce some modest measures — cracking down on a handful of banks or companies without potent political ties — as a way of giving Mr. Trump just enough evidence to show that his confidence in Mr. Xi was not wholly misplaced.
“There is a perception that the Chinese played to his vanity and fleeced him,” said Christopher R. Hill, a special envoy on North Korea during the George W. Bush administration. “It puts him in an awkward position. My sense is they know that calculation and want to deliver on it.”