BRUSSELS—The European Union’s top trade official arrived in Washington on Tuesday with a daunting task: convince the Trump administration that, contrary to White House claims, the bloc’s economic relationship with the U.S. is fair.

In two days of meetings, EU Trade Commissioner Cecilia Malmström will press counterparts including Commerce Secretary Wilbur Ross to exempt the bloc from President Donald Trump’s steel and aluminum tariffs, which are set to take effect Friday.

Mr. Trump invoked a threat to national security to justify the tariffs, leaving the EU aghast, given that 22 of the bloc’s 28 members are also North Atlantic Treaty Organization allies of the U.S.

EU officials are urging the White House to consider both significant trans-Atlantic economic links and risks the duties could pose to the broader alliance that shaped the postwar global order. The EU is also calling on the U.S. to jointly tackle China’s market-distorting policies instead of punishing partners world-wide.

“What we want to do is to clear up this mess,” Ms. Malmström told the European Parliament last week. “We don’t want a trade war.”