Mr. Trump has told aides that he believes that Democrats have the potential to appear overly partisan in investigating his family and that voters may be sympathetic to efforts to rebuff them.

Thomas M. Reynolds, a former congressman who was the chairman of the National Republican Congressional Committee in 2006, agreed that the incoming House majority would be wise not to meddle excessively in the private affairs of the president’s business and two adult sons.

But, he said, “we’re on a thin line here because the president still owns the company.”

Congressional inquiries are treacherous, and can be hard to weather. The president is said to be unprepared for the onslaught he is facing, even as the White House Counsel’s Office, which will be led by a new lead counsel, Pat Cipollone, is hiring a raft of lawyers.

Ms. Trump and her husband, Mr. Kushner, have had a veteran Washington lawyer, Abbe D. Lowell, working with them for several months. But Mr. Trump is famously frugal about hiring personal lawyers, and it is unclear whether he or his adult sons are prepared for how much assistance they will require.

“Politically speaking, this is the Achilles’ heel of the administration,” said Representative Jamie Raskin of Maryland, a Democrat who sits on two of the key investigative panels, referring to the Trump family’s business dealings. “They have turned the government of the United States into a moneymaking operation for the president and his family and close friends. That is the exact opposite of our constitutional design.”

Both Mr. Kushner and Ms. Trump were advised by White House lawyers of the risks of joining the West Wing as formal advisers two years ago, since it would open them up to official lines of inquiry, among other issues. On the other hand, people close to them noted that they might be shielded by executive privilege in some instances, whereas the president’s adult sons, who do not serve in the administration, are not.

Democrats have said they will not reflexively file subpoenas, but they have already gotten signals from the White House that cooperation is likely to be minimal. Even if the Mueller investigation ends in the coming months, as Mr. Trump’s lawyers believe is possible, the White House is facing a string of congressional subpoenas that could grind the gears of government to a halt, and put a spotlight on areas that the secretive Mr. Trump prefers to keep out of sight.