“There are many people in our country willing to reach across the divide, regardless of what you’ve heard in this campaign,” she said.

For Republicans, blunting Mrs. Clinton’s ability to carry other Democrats into office has become the overriding imperative in the final weeks of the 2016 race. With Mr. Trump so diminished as a competitor for Mrs. Clinton, Republicans say they will now ask voters in newly explicit terms to elect a divided government rather than giving Mrs. Clinton unchecked power.

The Congressional Leadership Fund, a powerful “super PAC” that supports Republicans in the House of Representatives, is to begin running ads in the coming days that attack Democratic candidates as “rubber stamps” for Mrs. Clinton and urge voters in swing districts to support Republicans instead.

Mike Shields, the group’s president, said it had tested that message and found it effective in closely contested races, even with voters who are likely to support Mrs. Clinton over Mr. Trump.

“There are many districts where we are going to be running ads that talk about the Democrat being a rubber stamp for Hillary Clinton,” Mr. Shields said. “In many districts, it is a very, very potent weapon to use against a Democratic candidate for Congress.”

Republicans fear Mr. Trump will do grievous damage to the party unless he can close the yawning gap with Mrs. Clinton in the presidential race. An ABC News tracking poll published on Sunday showed him trailing Mrs. Clinton by 12 percentage points nationally and drawing just 38 percent of the vote.

Mrs. Clinton, who drew support from 50 percent of voters in the poll, was openly dismissive of Mr. Trump over the weekend, telling reporters on Saturday that she no longer worried about answering his attacks. “I debated him for four and a half hours,” she said. “I don’t even think about responding to him anymore.”