One key bloc in the House, the Congressional Black Caucus, has taken no formal position, though some of its members are leading the charge on impeachment. “We’re probably going to wind up with no alternative but to move to impeachment, but I don’t think we are there today,” said the group’s chairwoman, Representative Karen Bass of California, who is planning an educational session on impeachment for members when they return to Washington.

In another sign of momentum in that direction, Representative James Clyburn of South Carolina, the No. 3 House Democrat, said Sunday that in his view it was more a question of when than if the House would hold impeachment proceedings.

“What I have said time and time again is, Mueller has developed the grounds for impeachment,” Mr. Clyburn said on CNN’s “State of the Union.” “The House has to determine the timing for impeachment. There’s a big difference.”

But he cautioned against moving too quickly, and the publicly declared supporters still fall well short of the 218 or so that would be required if Speaker Nancy Pelosi were to put it to a vote. More to the point, the backers of impeachment have not convinced the speaker that there is enough public support to warrant taking a step that will further cleave the country.

“Impeachment is a political act, and you cannot impeach a president if the American people will not support it,” Representative Jerrold Nadler of New York, the House Judiciary Committee chairman who leads the Democrats’ existing investigation into obstruction of justice and abuse of power, said during an appearance on Friday on WNYC.

But Mr. Nadler has privately made a case to Ms. Pelosi for opening an impeachment inquiry, and he suggested opinion could shift: “The American people right now do not support it because they do not know the story. They don’t know the facts. We have to get the facts out. We have to hold a series of hearings. We have to hold the investigations.”