Long before Eric Trump’s statements on Thursday, some senior executives at the Trump Organization expressed concern that expanding the hotel operation could backfire and reflect poorly on the president, according to the people briefed on the matter, who spoke on the condition of anonymity because they were not authorized to discuss it. The executives worried that any potential new deals could lead to accusations that the company — and its owner — were profiting from the presidency.

In late 2017, Eric Trump foreshadowed a shift in strategy, indicating in an interview that the company would concentrate on its existing golf, real estate and hotel properties.

“If we have to take a break for an eight-year period of time or a four-year period, then it is what it is,” he said, referring to new business opportunities.

At the outset of the presidency, Mr. Trump’s sons inherited a company that was coming off a decade of growth, having opened a slate of golf courses and luxury hotels from Chicago to Las Vegas. Marketing deals put the Trump name on properties around the world, as well as on suits, mattresses and a host of other products.

The presidency proved to be a game changer.

The Trump name became so toxic in some places that the company was paid to remove it from hotels in Toronto and New York. The majority owner of the Trump hotel in Panama took a more drastic step, ordering the T-R-U-M-P letters pried off the property with a crowbar.

In the last two years, the company has also been under scrutiny from federal investigators and Democrats in Congress, particularly after Mr. Cohen pleaded guilty to arranging hush money payments during the presidential campaign to two women who said they had had affairs with Mr. Trump. The Trump Organization reimbursed Mr. Cohen for one of those payments, and is itself a focus of the ongoing investigation.