Those calculations will be on display in Osaka, where Mr. Trump will join counterparts for the annual Group of 20 summit meeting hosted by Prime Minister Shinzo Abe of Japan. While in Osaka, Mr. Trump is planning sessions with Mr. Abe as well as with President Xi Jinping of China, President Vladimir V. Putin of Russia, Crown Prince Mohammed bin Salman of Saudi Arabia, Prime Minister Scott Morrison of Australia, Prime Minister Narendra Modi of India, Chancellor Angela Merkel of Germany and President Recep Tayyip Erdogan of Turkey.

Many of those meetings will be fraught for various reasons. Mr. Trump’s session with Mr. Xi will be closely watched by markets around the world to see if the two can make progress on resolving the multibillion-dollar trade war that has already led to escalating tariffs and threats of even more, with American farmers, importers and consumers caught in the crossfire.

His meeting with Mr. Putin will be their first since the special counsel, Robert S. Mueller III, issued his report documenting an expansive Russian operation to tilt the 2016 American election to Mr. Trump. While Mr. Mueller did not establish a criminal conspiracy between Mr. Trump and Russia, he outlined an extensive series of contacts between the two camps.

Likewise, Mr. Trump’s meeting with Prince Mohammed will be their first official session since American and United Nations agencies concluded that Saudi Arabia’s de facto ruler was almost certainly behind the grisly murder and bone saw dismemberment of Jamal Khashoggi, the Saudi dissident and journalist for The Washington Post. Mr. Trump said last week that he would not let that episode disrupt the relationship because of lucrative arms sales to the Saudis.

As if those encounters were not delicate enough, after the G-20 conference Mr. Trump will fly to Seoul, South Korea, where he will meet with President Moon Jae-in to discuss ways of re-energizing a stalled diplomatic venture with Kim Jong-un, the leader of nuclear-armed North Korea. Mr. Trump and Mr. Kim recently exchanged friendly letters after months of silence following their failed nuclear summit meeting in Hanoi, Vietnam, in February.

After two and a half years of his presidency, “most of the leaders around the table are learning to adapt to Trump’s erratic blend of unilateralism, impulsiveness and narcissism,” said William J. Burns, a career diplomat and former deputy secretary of state who is now the president of the Carnegie Endowment for International Peace.

Mr. Burns, the author of the recent memoir “The Back Channel,” said leaders were not only “uncertain at this stage about the outcome of the 2020 U.S. presidential election,” but they were also “increasingly uncertain about whether the drift in America’s role is personal — and idiosyncratic to Trump — or structural.”