President Trump’s tariff threats have turned into a play for cold, hard cash as he tries to leverage U.S. economic power to cajole other nations to make multibillion-dollar investments in order to maintain access to America’s market.

The president’s second-term trade agenda has clear echoes of his “Art of the Deal” approach, essentially demanding that trading partners show him the money in the form of investment pledges or else face astronomically high tariffs.

The financial promises give Mr. Trump the opportunity to flex his negotiating prowess in relatable terms and show off the splashy sums he is pulling into America, adding to the reality show intrigue of his trade agenda. As the Trump administration races to reach trade deals with dozens of countries ahead of a Thursday deadline, he has embraced a strategy that goes beyond opening international markets and reducing the U.S. trade deficit.

The tactic was on display last week as Mr. Trump and his team rolled out a blitz of new trade agreements before a self-imposed Aug. 1 deadline.

“South Korea is right now at a 25% Tariff, but they have an offer to buy down those Tariffs,” Mr. Trump wrote on social media on Wednesday. “I will be interested in hearing what that offer is.”

The next day, Mr. Trump agreed to impose a tariff of 15 percent on imports from South Korea. The lower rate came after South Korea agreed to make $350 billion in investments in the United States and purchase $100 billion of liquefied natural gas.