Mr. Viola is not the only appointee with a complicated portfolio.

On Wednesday, Anthony Scaramucci, a former Trump campaign fund-raiser who was awaiting swearing in as the White House liaison to the business community, was told to step aside in the face of concerns over the recent sale of his investment firm to a Chinese company with deep ties to the nation’s Communist Party.

And in January, The New York Times reported that Jared Kushner, Mr. Trump’s son-in-law, had been involved in negotiations to sell a stake in his family’s flagship building to a large, opaque Chinese company.

Mr. Kushner, who was subsequently appointed a senior adviser to the president, has resigned as chief executive of his family’s real estate firm and announced plans to sell various assets to his brother or to a family trust controlled by his mother, a plan that some ethics experts said did not go far enough.

In Mr. Viola’s case, the current status of the negotiations for his airline, which have not been previously reported, and the financial terms under discussion are not known. Eastern Air Lines — the latest, and far smaller incarnation of the once-famous domestic airline — is worth upward of $25 million and is 75 percent owned by Mr. Viola, according to one person with knowledge of its operations, who spoke on the condition of anonymity.