House Democrats, set to take over majority control next year, plan to probe President Trump’s involvement in hush payments arranged during the 2016 campaign to two women alleging affairs, according to a Democratic aide on the House Oversight Committee.

The aide said House Democrats have already begun probing the hush payments, including requesting records from the Trump Organization in September, and “plan to dig in deeper” by examining Mr. Trump’s role.

The Wall Street Journal on Friday reported that Mr. Trump was involved in or briefed on nearly every step of the agreements arranged by his former lawyer, Michael Cohen. That contradicted two years of denials by the president, his advisers and his legal team that Mr. Trump had any knowledge of the deals at the time they were struck—and raised the possibility that the president violated federal campaign-finance laws.

The White House referred a request for comment on House Democrats’ planned inquiry to the president’s outside legal team, which didn’t respond.

The Journal’s account was based on interviews with three dozen people directly involved in or briefed on the events. Among the findings: David Pecker, chief executive of American Media Inc., promised Mr. Trump during an August 2015 meeting at Trump Tower to buy the silence of women who might allege during the campaign that they had past sexual encounters with the Republican candidate. In June 2016, Mr. Trump asked Mr. Pecker, whose company publishes the National Enquirer, to quash the story of a former Playboy model, Karen McDougal, who said they had had an affair in 2006.

Mr. Cohen told federal prosecutors in Manhattan that in October 2016, he informed Mr. Trump that he needed to find a way to pay Stephanie Clifford, the former adult-film star known as Stormy Daniels, to keep her from publicly discussing an alleged sexual encounter with Mr. Trump, also in 2006. Mr. Trump told Mr. Cohen in October 2016 to “get it done,” according to Mr. Cohen’s account to prosecutors.

While President Trump publicly fought with women leading up to the 2016 election, in private he directed schemes to silence their stories of two alleged affairs. Here’s a timeline of Trump’s personal involvement.

Mr. Trump has denied having had sex with either Ms. McDougal or Ms. Clifford.

Mr. Trump and his business dealings already are the subject of multiple investigations. Manhattan federal prosecutors who investigated Mr. Cohen are now examining business dealings by the Trump Organization, according to people familiar with the matter. The Trump Organization declined to comment on the investigation.

Special counsel Robert Mueller, who separately has been investigating whether Trump associates colluded with Russia’s efforts to interfere in the 2016 U.S. election, is expected to file a report on his findings to the Justice Department. Mr. Trump has denied collusion, and Moscow has denied interfering in the election.

An inquiry by House Democrats—whose majority status next year will mean they gain subpoena power—would open up a new avenue of investigation into whether Mr. Trump committed campaign-finance violations during the 2016 election.

Rep. Jerry Nadler (D., N.Y.), who is a favorite to lead the House Judiciary Committee next year, told CNN Sunday that if the president was involved in criminal campaign-finance violations, “that might very well be an impeachable offense.”

Asked about the Journal’s report, Mr. Nadler said it would be a question of whether Mr. Trump’s involvement could be proved and “whether the situation is serious enough that it makes sense to do an impeachment to defend the system of government and the system of democracy.”

As chairman of the Judiciary Committee, Mr. Nadler would preside over impeachment hearings if they were to start.

The White House and the Trump Organization have previously rebuffed requests made by Democrats in Congress that haven’t been co-signed by majority Republicans, though the company has responded to bipartisan requests from lawmakers.

Rep. Elijah Cummings of Maryland, the top Democrat on the House Oversight Committee, in September sent a letter to then-White House Counsel Don McGahn and to the Trump Organization requesting documents related to Mr. Trump’s failure to disclose his debts to his former lawyer, Mr. Cohen, on a financial disclosure last year.

The request came soon after Mr. Cohen pleaded guilty in Manhattan federal court to campaign-finance violations in connection with the payments to Ms. McDougal and Ms. Clifford and told a judge in open court that he had been directed by and coordinated with Mr. Trump. Mr. Cohen, who also pleaded guilty to tax evasion and making false statements to a bank, is scheduled to be sentenced on Dec. 12.

The Oversight Committee didn’t receive a response from the White House or the Trump Organization, according to the Democratic aide.

In May, Mr. Trump filed a financial disclosure with a footnote documenting a reimbursement of between $100,001 and $250,000 for expenses in 2016 to Mr. Cohen for payment to Ms. Clifford. The payment wasn’t listed in his previous financial disclosure, released last year. Disclosure rules require liabilities to be listed.

—Michael Rothfeld contributed to this article.

Write to Rebecca Ballhaus at Rebecca.Ballhaus@wsj.com