Other Republicans dispute any suggestion that Mr. Cruz could lose. Texas, they argue, is not going to send a Democrat to the Senate. Some of its biggest cities are largely Democratic, but the suburbs and rural areas remain Republican, and Democratic voters’ history of low turnout has helped Republicans maintain their grip on power. In the Republican presidential primary in 2016, Mr. Cruz easily won Texas, earning 43 percent of the vote compared to Mr. Trump’s 26 percent.

“Ted Cruz is not in trouble,” said Allen E. Blakemore, a Republican political consultant and the senior strategist for one of the most powerful conservatives in the state, Lt. Gov. Dan Patrick. “Texas is a reliably red state and is going to continue to be for the foreseeable future. There is not a single statewide race that is in jeopardy.”

If Mr. Cruz is in the fight of his political life, it was hard to see on Tuesday.

“I didn’t give it much thought,” he said Tuesday when asked about Mr. Mulvaney’s comments. “I’m focused on what the people of Texas say. And we’re seeing incredible energy, incredible enthusiasm.” Mr. Cruz said he has been approached by voters who say they are Democrats but are supporting Mr. Cruz now.

“With the election of Donald Trump, national Democrats are just consumed by rage and fury, and they’re embracing extreme positions, like Beto O’Rourke saying he’s open to abolishing ICE, or Beto O’Rourke committing that he would vote ‘yes’ to impeach Donald Trump,” Mr. Cruz said. “I am energetically welcoming conservative Democrats, moderate Democrats, independents.”

Mr. Cruz’s campaign is getting a boost soon from Mr. Trump, his rival for the Republican presidential nomination in 2016. Mr. Trump announced on Twitter that he was coming to Texas in October to host a “major rally” for Mr. Cruz. Democrats see it as a sign that Mr. Cruz’s campaign is struggling and needs rescuing.

The trip seemed, in part, to be a response to a meeting in July in Washington between Texas leaders and Mr. Trump’s political staff. The July 25 meeting at the Eisenhower Executive Office Building was attended by Mr. Patrick, who was the Texas chairman of Mr. Trump’s campaign in 2016. Mr. Patrick and others were seeking to encourage Mr. Trump to campaign in Texas, according to a person who attended the meeting, who spoke on condition of anonymity because he was not authorized to discuss the meeting. The idea was not for Mr. Trump to come to the aid of Mr. Cruz, but for the president to boost Republican turnout to help the races farther down the ballot, the person said.

On the surface, the prospect of Mr. Cruz campaigning with the president seems remarkable. Mr. Trump nicknamed Mr. Cruz Lyin’ Ted during the 2016 campaign, and attacked both his wife and his father. Mr. Cruz called Mr. Trump a “pathological liar” and “a sniveling coward.”