The findings of the special counsel, Robert S. Mueller III, will significantly influence how they proceed, top House Democrats cautioned, but at the moment their preparations for battle are focused almost exclusively on unearthing new facts and challenging the Justice Department for access to the evidence that Mr. Mueller has gathered, not on meting out punishment.

Democratic pollsters who once helped assemble guidance for talking about the importance of the special counsel investigation are developing new polling to help lawmakers reframe the debate on presidential wrongdoing around Mr. Mueller’s possible conclusion, to lay the groundwork for continued investigations and to push back against the attempts by the president’s allies to play down the significance of wrongdoing.

“It should be the last thing on members’ minds,” Representative Mike Quigley, an Illinois Democrat who sits on the Intelligence Committee, said about impeachment. “You have to be patient and you have to be practical. You don’t stop a criminal investigation when you have enough. You stop when you have learned everything there is to be learned.”

The resistance to impeachment puts the leadership at odds with some rank-and-file Democratic lawmakers, liberal activists and a significant portion of Democratic voters who believe that sufficient evidence already exists to start the process of ousting the president. As House inquiries proceed, pressure will mount on Ms. Pelosi and other top Democrats to take action against Mr. Trump.

“There are only two questions left: How deep is the corruption and what do you want to do about it?” said Tom Steyer, the California billionaire who has crusaded for Mr. Trump’s impeachment and poured millions into organizing for town halls and national advertising to pressure Democratic lawmakers.