On July 22, Ryosei Akazawa, then Japan’s minister in charge of tariff negotiations with the United States, was sitting across from President Trump in the Oval Office. Mr. Trump, deploying a familiar negotiating style, began crossing out numbers on a placemat-size visual aid.

After eight trips to Washington, Mr. Akazawa had worked out the outline of a plan with Mr. Trump’s trade officials. But in the final meeting, the president upped the ante. Mr. Akazawa went along with the new demands.

Japan walked away with lower-than-threatened tariffs. In return, it pledged to put up $550 billion to fund American projects, an amount equal to more than a tenth of Japan’s economy. In an extraordinary provision, the deal gave the Trump administration the power to, in effect, dictate how the money is spent.

Three months later, both countries are still trying to figure out what those investments will be. A top concern for many Japanese officials is that the Trump administration may select projects that don’t make strategic sense for Japan. That would force a difficult choice: acquiesce, or refuse and face the risk of higher tariffs.

That will be the challenging backdrop when Sanae Takaichi, Japan’s new prime minister, meets with Mr. Trump on Tuesday. Their discussion is expected to focus on how her debt-strapped nation can both invest hundreds of billions of dollars in the United States and increase military spending — another move Washington is pressing for.

Heading into the storm, there is a constant: Mr. Akazawa, the “tariff minister” who ingratiated himself to the Trump administration. Mr. Akazawa has told people close to him that he tried to disarm Mr. Trump by being intentionally deferential. For a meeting in the Oval Office, he donned a “Make America Great Again” cap. He held numerous late-night phone calls with Commerce Secretary Howard Lutnick, and trekked to Washington nearly a dozen times.