“We’re going to try to work on issues with our Republican colleagues, even with the president, when we can,” said Senator Chuck Schumer of New York, the Democratic leader. “We believe we need to do something about guns. We believe we need to reduce the cost of prescription drugs. The $64,000 question is will the president want to work with us, given what happened. We shall see.”

The prospect of a compromise on new control gun laws could be the first casualty of the impeachment clash. The White House had already delayed the rollout of its awaited proposal, and administration officials have appeared torn on how far to go in responding to a spree of mass shootings in August.

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Getting any agreement on the highly charged issue was always going to be tough, even in a cooperative environment. Mr. Trump, angry over Ms. Pelosi’s new embrace of the impeachment inquiry, dismissed her on Wednesday, saying that she was no longer speaker in his mind and that she was “not interested in guns and protection and gun safety.”

Yet at the same time, Senator Christopher S. Murphy of Connecticut, a leading Democratic voice on gun issues, said that a White House official had reached out to him on Tuesday night — just hours after Ms. Pelosi announced to the world that Mr. Trump’s conduct had been a “betrayal” and the House would pursue an impeachment inquiry because of it — to express hope that gun control negotiations could continue in the hope of sealing a deal.

“I understand that will be very difficult, but I will take their outreach last night as sincere,” Mr. Murphy told reporters on Wednesday.

Efforts to lower prescription drug costs are one area where Mr. Trump and House Democrats seemed to have more in common with one another than Senate Republicans. Both sides are advancing competing proposals, and one influential senator warned that it would be a major opportunity missed if the effort to reduce prices was lost in the impeachment dispute.

“With only a finite number of legislating weeks left in the year, it is absolutely critical that lawmakers capitalize on the unprecedented momentum and act now to take up, advance and pass these market-based solutions to hold Big Pharma accountable and lower prescription drug prices,” said Senator Charles E. Grassley, Republican of Iowa and the chairman of the Senate Finance Committee.