Donald J. Trump’s support has plunged across the swing-state map over the last 10 days, wiping out his political recovery from September and threatening to undo weeks of Republican gains in the battle for control of Congress.

For his party, Mr. Trump’s reversal in fortune comes at the worst possible moment: Having muted their criticism of Mr. Trump in hopes that he could at least run competitively through Election Day, Republicans must decide in the next few days, rather than weeks, whether to seek distance from his wobbly campaign.

Should Mr. Trump falter badly in his second debate with Hillary Clinton on Sunday in St. Louis, Republican congressional candidates may take it as a cue to flee openly from their nominee, said two senior Republicans involved at high levels of the campaign who spoke on the condition of anonymity because they were not authorized to discuss private party strategy.

Mr. Trump has already slipped perceptibly in public polls, trailing widely this week in Pennsylvania and by smaller margins in Florida and North Carolina — three states he cannot afford to lose. But private polling by both parties shows an even more precipitous drop, especially among independent voters, moderate Republicans and women, according to a dozen strategists from both parties who spoke on the condition of anonymity because the data was confidential.