President Trump has successfully stiff-armed Congress’ attempts to probe his continued ties to his business empire even as he sits in the White House — until last week.

Mr. Trump appears to have crossed new lines after an aide to Vice President Mike Pence said the president personally suggested Mr. Pence and his entourage stay at one of the president’s properties in Ireland, on taxpayers’ dime.

Now that trip has become the subject of a congressional investigation, giving the president’s fiercest critics a new avenue as they ponder impeaching him.

The cost of the several-day stay at Doonbeg is not yet known, though congressional investigators say a similar trip by Mr. Trump and his family in June cost $3.6 million.

“The committee does not believe that U.S. taxpayer funds should be used to personally enrich President Trump, his family, and his companies,” House Oversight Committee Chairman Elijah E. Cummings, Maryland Democrat, said in letters to the White House, vice president, Secret Service and the Trump Organization.

The Irish adventure now joins another probe into the president’s promotion of Doral, his Miami resort property, as the site of next year’s summit of the Group of 7 nations.

Any violations of the Constitution’s “Emoluments Clause” — a ban on profiting from the presidency — “are of significant interest and grave concern to the committee as it considers whether to recommend articles of impeachment,” Judiciary Committee Chairman Jerrold Nadler, New York Democrat, wrote in his own letters to White House counsel and James M. Murray, the director of the Secret Service.

The Trump team was already scrambling last week to figure out a story that could protect Mr. Trump from legal jeopardy, reversing the original version that the president suggested the stay. Now Mr. Trump says the decision was Mr. Pence’s alone.

The Doral and Doonbeg affairs could spark momentum for Democrats looking to ding the president or build the case for impeachment ahead of 2020, said Jim Manley, a Democratic strategist.

“It takes time to get the legal process in order, but now they are ready to go — and it’s going to be a long fall for the administration,” said Mr. Manley, who served as a spokesman for former Senate Democratic Leader Harry Reid of Nevada.

Congress is also looking at a March incident, first reported by Politico, in which the crew of a U.S. Air Force C-17 stayed at Mr. Trump’s Turnberry golf resort in Scotland amid a refueling stop on the way to Kuwait. The Air Force said in a statement said the stop was “not unusual,” though it’s another instance in which taxpayer money appeared to flow to Mr. Trump’s personal company.

Mr. Trump has maintained ownership of his business empire after becoming president, though he turned over day-to-day operations to his children.

Investigations by press outlets have found foreign governments have poured money into his properties since he took office. And even Attorney General William P. Barr has scheduled his annual Christmas party for a Trump hotel in Washington this year.

Those sorts of moves had spawned lawsuits arguing the payments were emoluments, violating the Constitution.

Mr. Trump had been winning some of the legal skirmishes, getting one lawsuit brought by Maryland and the District of Columbia tossed this summer. And the president’s team has managed to slow down another lawsuit by winning the right to appeal at an early stage.

Mr. Trump didn’t return fire at the Democratic chairmen over the weekend, though he’s bristled at attacks on his properties, saying he’s on track to lose money while serving as president because he has to forgo big deals.

The Doonbeg affair put him on the back foot, however, with Mr. Cummings saying the shifting explanations out of Mr. Trump’s team add to the fishy stench.

“The White House has not made public how much vice president’s trip cost the American taxpayer — or benefited the Trump Organization — but based on previous investigations by the Government Accountability Office, the bill could be significant,” Mr. Cummings wrote to White House Chief of Staff Mick Mulvaney, Trump Organization Chief Financial Officer Allen Weisselberg and others.

Mr. Nadler, meanwhile, says if the president follows through on his plan to host the next summit at his Florida property, it would be the “first publicly known instance” in which foreign governments would be required to spend money at Trump-branded properties to engage in diplomacy with the U.S.

Mr. Trump called his Miami resort a great fit when he made the suggestion during the most the recent G-7 summit in France. He said White House scouts couldn’t find a better site for next year’s meetings.

“It’s, like, such a natural,” Mr. Trump said. “Each country can have their own villa or their own bungalow.”

Mr. Nadler wasn’t impressed.

“The threat that the president’s personal financial interests could shape decisions concerning official U.S. government activities,” he wrote, “is precisely the type of risk that the emoluments clauses were intended to minimize.”

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