“No, not fear-mongering at all,” Mr. Trump shot back when asked about his attempts in recent days to stoke alarm about an immigrant onslaught. “Immigration is a very important subject.”

At the same time, with Republicans privately grim about their chances of holding the House, Mr. Trump was bullish about the party’s chances, and argued that Republicans were doing particularly well with women, who polls show are particularly hostile to him and the party overall.

“Women, we’re doing very well with the women vote, because they want security,” Mr. Trump said. “They want safety. They don’t want these people pouring into our country, totally unchecked.”

The president later said he would not blame Mr. Ryan if the party lost control of the House next week, but he was quick to declare himself a net positive for Republicans, who he argued had gotten a boost from his campaign stops.

“No, I’m not going to blame anybody,” Mr. Trump said when asked about Mr. Ryan. “I think we’re doing well with the House. We’re going to have to see. It’s a lot of people. I’ve campaigned for a lot of candidates that were down a little bit, and now they’re up.”

Yet the president’s antics have also introduced an unwelcome element of unpredictability for Republican candidates in the final days of competitive races. Several Republicans running in suburban districts, more politically moderate districts or those with large concentrations of Latino voters quickly disavowed the idea of ending birthright citizenship, relishing a chance to highlight a topic of disagreement with a president who is not popular with the voters whose support they need to win.

“I can see a strategy on the president’s part of hitting the immigration and border security issues hard, even if the facts don’t support his positions, in order to excite his base,” said Michael Steel, a Republican strategist who is a former senior aide to Mr. Ryan. But, he added, “picking a fight with any political figure in the party with less than a week to go before the midterms is just stupendously counterproductive, and implies either a lack of discipline and thin-skinnedness, or that the president is bracing for the blame game in case of a bad result.”