BRUSSELS — Gordon D. Sondland, the blunt-spoken hotelier who is President Trump’s ambassador to the European Union, was boasting on Ukrainian television that Mr. Trump had honored him with a “special assignment” — “overseeing” relations between the two countries “at the highest levels.”

Mr. Sondland had arrived in Kiev on July 25, the day of the now-infamous telephone call between President Trump and the new president of Ukraine, Volodymyr Zelensky. He had spoken to Mr. Trump minutes before the call, he said, and met with Mr. Zelensky for an hour the next morning, before his television interview.

At the time, his television remarks might have been the kind of diplomatic bluster one would expect of Mr. Sondland, a big, loquacious man who has been a prominent Republican donor and fund-raiser for years and loves to remind people of the good relationship he has developed with Mr. Trump.

But in the glare of the impeachment inquiry swirling in Washington, Mr. Sondland’s mission is now being scrutinized in an entirely different light, to assess whether it was to give a lift to American relations with Ukraine, or actually to serve as Mr. Trump’s personal fixer.