With less than a week left for dozens of countries to negotiate trade deals with the Trump administration to avoid tariffs, some of America’s trading partners are in last-minute negotiations with the United States to stave off levies that could escalate a broader trade war.

One of the biggest meetings came in Scotland on Sunday, when President Trump met for trade talks with Ursula von der Leyen, the president of the European Commission, the European Union’s executive arm. Both parties agreed to a trade deal that sets a 15 percent tariff on most European goods, potentially avoiding a trade war that could have been damaging for both economies.

Ms. von der Leyen called it “the biggest trade deal ever,” while Mr. Trump said, “This is the biggest of them all.”

The European Union is, by some measures, America’s single most important trading partner, a bloc of 27 nations that collectively make up one of the world’s largest economies.

There are still many details left to work out after Sunday’s preliminary agreement. Separate negotiations are continuing this week on major details of the deal, including steel and aluminum tariffs.

Reactions in Europe have been varied, with some leaders voicing cautious support, while others called the deal unsatisfactory and lopsided.