WASHINGTON — He likes one and admits to falling in love with the other. He treasures letters from each of them. And he believes the personal rapport he built with both could clear the way to historic agreements on trade and nuclear arms that have eluded his predecessors.

To President Trump, all diplomacy is personal, especially with the two Asian strongmen he has courted most avidly, Kim Jong-un of North Korea and Xi Jinping of China.

But Mr. Trump’s honeymoon with Mr. Kim came to an abrupt end in Vietnam last month, and his conviction that agreements between nations are little different than real estate deals between bosses now faces another stiff test with the Chinese president.

In trade talks with China that are heading toward a potential meeting with Mr. Xi, Mr. Trump hopes to overcome decades of distrust to sign an agreement that would end a bitter trade war between the world’s two largest economies.