Democrats have been hesitant to loudly advertise the specifics of the potential investigative blitz, convinced that swing voters are more likely to back them based on kitchen-table economic issues like wages, health care and retirement benefits. Loyal Democratic voters do not need to be reminded that a Democratic House will check the power of the president, they argue.

“If this is a referendum on Trump, the way I would want to frame it is not ‘remove or retain’ but ‘contain or enable,’” said Representative Brad Sherman, Democrat of California, who has already introduced an article of impeachment against the president. “There are more votes for ‘contain’ than there are for ‘remove.’”

But with Mr. Trump’s longtime personal lawyer, Michael D. Cohen, implicating the president directly in the payoffs to Stephanie Clifford and Karen McDougal, the conviction of Mr. Trump’s former campaign chairman in federal court and a rash of indictments and other alleged wrongdoings swirling around House Republicans themselves, the Democrats are increasingly selling themselves as a much-needed antidote to a “culture of corruption” in the capital.

Democrats believe the Republicans abused the power of the majority to hobble the Obama administration, deeply damage Hillary Clinton and protect Mr. Trump. That frustration, coupled with what most lawmakers expect to be a wave of Democratic anti-Trump outrage fueling midterm victories, could overwhelm the instincts of more moderate members of the party to chart a different, more bipartisan course than Republicans have.

Democrats on the Oversight Committee, typically the House’s most muscular investigative body, have more than 50 subpoena requests that have been denied by committee Republicans since Mr. Trump took office, from the administration of security clearances at the White House to chartered jet travel by cabinet officials to Justice Department documents related to its decision not to defend the Affordable Care Act in court.

“It’s not like we have to go dig them up. They are right there sitting on the desk,” Mr. Cummings said.