But Speaker Nancy Pelosi remains skeptical, telling colleagues during a private call late last month that the public still “isn’t there on impeachment.” Many of the caucus’s more moderate members, whose districts are crucial to maintaining the Democrats’ majority, have not backed impeachment either. And Republicans remain unified behind Mr. Trump.

Just as consequentially, court cases have hamstrung Democrats’ ability to stage potentially powerful public hearings — in part as a result of Mr. Trump’s stonewalling of congressional oversight efforts. Rulings in two cases — one on unsealing grand jury secrets from Robert S. Mueller III’s investigation, and the other to enforce a subpoena for the former White House counsel Donald F. McGahn II — are expected this fall. They could determine whether lawmakers will be able to blunt the White House’s attempts to run out the clock by slow-walking document production and ordering major witnesses not to appear before lawmakers without a court order.

For now, Democratic congressional investigators agree they should push ahead, even if impeachment ultimately remains beyond their grasp.

“If we aren’t able to collect the evidence that we need to present a credible case before the election, well, at least maybe we will have put enough evidence out there that the public can exercise another form of regime change that is in the Constitution and vote,” said Representative Mary Gay Scanlon, a Pennsylvania Democrat on the committee.

The Judiciary Committee will take a substantial step to organize its effort this week. Lawmakers are expected to vote to establish rules and procedures governing the inquiry, including allowing staff lawyers to question witnesses and the president’s lawyers to more formally offer a defense, according to an official familiar with the committee’s plans.

Democrats began examining the hush payments and other areas of scrutiny in the spring, requesting documents and taking other early steps. But they have focused mostly on Mr. Mueller’s investigation and his account of Mr. Trump’s repeated attempts to thwart his team.