On her way out of town last week, Ms. Pelosi blessed a proposal by the Judiciary Committee to take the position in court that the panel had already begun, on its own authority, “investigating whether to recommend articles of impeachment” against the president. Therefore, the panel said, Democrats did not actually need a House vote of the sort that was taken to initiate impeachment inquiries into Presidents Richard M. Nixon and Bill Clinton.

“The stance that she has taken is going to stay put for awhile,” said Representative Mike Quigley, Democrat of Illinois and an advocate of an inquiry.

Democratic leaders always recognized that the August break could be an inflection point for some lawmakers, when conversations with their constituents could push them toward endorsing impeachment. But the drive toward an inquiry seems to be driven as much by internal politics on Capitol Hill as any push from voters. Impeachment was barely a whisper in two nights of Democratic presidential primary debate.

Mr. Quigley said individual members’ views are being shaped by a range of factors, including possible primary challenges, Mr. Mueller’s testimony last week, comments by Mr. Trump that are widely condemned as racist and the administration’s refusal to comply with certain investigative requests by Congress.

In June, Mr. Engel picked up a Democratic primary challenger, Jamaal Bowman, a middle school principal from the Bronx with the backing of Justice Democrats, the insurgent group that helped lift Alexandria Ocasio-Cortez to victory in her primary campaign against longtime Representative Joseph Crowley.

A group of Democrats from Washington State, along with Ms. Murray, made the jump in favor of an inquiry together on Sunday, saying that they had waited to hear directly from Mr. Mueller and that their decision should come regardless of politics.