President Trump’s decision to impose tariffs on imported steel and aluminum has prompted a stampede by foreign countries and companies and their American partners pressing for exemptions and exclusions that could be worth billions of dollars in trade.

Corporations and foreign leaders are leaning on personal relationships in vying for meetings with White House officials, hiring lawyers and lobbyists to defend them, and drafting messages to persuade the public of the importance of free trade.

On Thursday afternoon, top oil and gas executives left an American Petroleum Institute board meeting at the Trump International Hotel in Washington and headed to the White House to personally lobby Mr. Trump, telling him that his pending steel tariffs could raise the cost of pipelines and extraction projects across the country, and discussing possible exemptions.