WASHINGTON — When two Republican members of Congress began formally questioning last week Ivanka Trump’s use of private email for government business, it was seen by people close to the White House as a sign of things to come for the president’s family.

One of the Republicans was Sen. Ron Johnson of Wisconsin, chairman of the Senate Homeland Security and Governmental Affairs Committee, which has conducted little oversight of the Trump White House until now.

The other was Rep. Trey Gowdy of South Carolina, chairman of the House Oversight Committee, who previously led a two-year investigation into events surrounding the attack on U.S. diplomatic outposts in Benghazi, Libya, focusing relentlessly on the role of Hillary Clinton. His most prominent investigation as chairman has scrutinized alleged anti-Trump political bias within the FBI during its inquiries related to the 2016 presidential campaign.

“That you now have Republicans investigating members of the first family is an indication of the perils ahead” for the Trumps, said Tom Davis, the former House Republican from Virginia who was chairman of the Oversight Committee from 2003 through 2006.

Gowdy, who is retiring from Congress in January, will have little to do with any investigation, and his role in endorsing the inquiry was seen as pro forma. In his place as chairman of the committee will be Rep. Elijah Cummings, D-Md., one of the newly empowered House Democrats eager to take on the Trump administration.

The Democrats are already laying out lines of inquiry that could quickly lead not just to Trump and his White House aides, but also to his immediate family. And Republicans returning to Capitol Hill next year may be forced by the changed political climate to take a harder line toward the Trump family.

Likely Democratic targets include not only the president’s personal finances and those of the Trump Organization, but also the actions taken by his sons Donald Jr. and Eric and his son-in-law, Jared Kushner, during the 2016 campaign and its aftermath.

The Oversight and Judiciary committees are likely to focus on any violations of the Constitution’s emoluments clause, which bans payments from foreign governments to federal officeholders, an avenue that will lead to an examination of payments to Trump-held properties and the role of his sons and Kushner. And some investigators may try to question Donald Trump Jr. again about a meeting he held with a Kremlin-linked lawyer promising “dirt” about Clinton.

While some of the areas primed for inquiry — most notably, ties between the Trump campaign, the Trump Organization and Russia, and possible campaign finance violations — are likely to overlap with investigations by special counsel Robert Mueller and federal prosecutors in New York, Democrats believe they have a broader mandate to question everything from foreign business dealings by Ivanka Trump and the Trump Organization to the administration of security clearances at the White House, including the one given to Kushner.

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 faces, 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.

Ivanka Trump and her husband, Kushner, have had a veteran Washington lawyer, Abbe D. Lowell, working with them for several months. But 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 Rep. Jamie Raskin, D-Md., 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.”

Raskin added: “The problems run from the constitutional to the statutory to the ethical to the contractual, and we on the Oversight Committee will have our hands full trying to keep up.” Kushner and Ivanka 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 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 Trump prefers to keep out of sight.

The president’s tax returns, documents he has steadfastly refused to release in defiance of long-standing traditions, remain a central target.

Democrats on the Ways and Means Committee are preparing to use a century-old provision in the federal tax code to try to gain access to Trump’s tax returns. Aides working for the committee’s soon-to-be chairman, Rep. Richard E. Neal, D-Mass., believe that Trump administration officials at the Treasury Department could refuse the request, forcing the battle into court.

But if Democrats are successful, the returns could offer them their first comprehensive look into the heart of the Trump Organization and the finances of the first family. The returns could also get wrapped into the Oversight and Judiciary Committees’ looks at potential violations of the emoluments clause.

Democrats have already brought a federal lawsuit contending that Trump’s financial interest in the Trump International Hotel in downtown Washington is such a violation. But congressional officials said they were interested more broadly in the possibility that the Trump family is profiting off their time in office, through, among other things, trademarks granted to the Trump Organization and Ivanka Trump’s business abroad.

Rep. Adam Schiff, D-Calif., who will be the chairman of the Intelligence Committee, has said that as he weighs how to reopen the committee’s investigation of Russian election interference, he intends to zero in on possible money laundering by Russians through the Trump Organization.

That line of inquiry may overlap with the federal investigations, but it would most likely involve an attempt to obtain the president’s bank and other business records.

Schiff has repeatedly criticized Republicans for refusing to demand the phone records of Donald Trump Jr., who exchanged a series of phone calls in the run-up to a 2016 Trump Tower meeting with a Russian lawyer in which he expected to receive dirt on Clinton, including with a blocked number that Democrats believe could have belonged to his father.

Rep. Eric Swalwell, D-Calif., who sits on both the Intelligence and Judiciary committees, said Democrats would seek to avoid “parlor intrigue and voyeuristic interests” in favor of potential irregularities shaping government policy, like Trump’s foreign financial dealings.

“Yeah, it is pretty objectionable that the president’s son-in-law failed a number of times to properly fill out a security clearance form, and it is objectionable he still has one now,” Swalwell said. “But is it the most objectionable thing the Trump administration has done? Not by a long shot.”