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Senator Ted Cruz of Texas, who has carefully avoided public criticism of Donald J. Trump, said this week at a private fund-raiser that the real estate developer was facing a “challenging question” about whether he has the “judgment” to be president.

Mr. Cruz, a rival of Mr. Trump’s for the Republican presidential nomination, is trying to position himself to be the beneficiary of any erosion of support for Mr. Trump, who is handily beating the rest of the field in opinion polls.

While he has said he does not agree with Mr. Trump’s latest provocative proposal, a restriction on Muslim immigrants entering the United States, he has taken pains to praise Mr. Trump for making immigration a focal point of his candidacy.

But inside a conference room Wednesday in a Madison Avenue office, with about 70 people pressed around a table, Mr. Cruz gave his assessment of the race, lumping Mr. Trump with another candidate whose supporters the Texas senator hopes to poach, Ben Carson.

“Both of them I like and respect,” said Mr. Cruz, according to an audio recording of his comments provided by one attendee. “I don’t believe either one of them is going to be our president.”

But he added, “You look at Paris, you look at San Bernardino, it’s given a seriousness to this race, that people are looking for: Who is prepared to be a commander in chief? Who understands the threats we face?”

He went on: “Who am I comfortable having their finger on the button? Now that’s a question of strength, but it’s also a question of judgment. And I think that is a question that is a challenging question for both of them.”

On the audio, after he described the “challenging question” facing Mr. Trump and Mr. Carson, Mr. Cruz went on to explain: “So my approach, much to the frustration of the media, has been to bear hug both of them, and smother them with love.”

He added: “People run as who they are. I believe gravity will bring both of those campaigns down” and “the lion’s share of their supporters come to us.”

Mr. Cruz later said he was “thrilled” that Mr. Trump was in the race because “he has framed the central narrative of this primary as who will stand up to Washington” and said he saw his own record as one of having “stood up to Washington.” After Mr. Cruz spoke, one person at the fund-raiser questioned Mr. Trump’s having “his finger on the Supreme Court button,” according to the attendees, a reference to the president’s ability to nominate the justices.

Hope Hicks, a spokeswoman for Mr. Trump, declined to comment about Mr. Cruz’s remarks on Thursday.

Mr. Cruz, questioned about the remarks on Thursday after a speech at the Heritage Foundation in Washington, said he was “not going to comment on what I may or may not have said at a private fund-raiser.”

“What I will say is this,” he continued. “In the course of a presidential election, the voters are going to make a decision about every candidate. And ultimately the decision is, who has the right judgment — experience and judgment — to serve as commander in chief? And every one of us who is running is being assessed by the voters under that metric. And that is exactly why we have a democratic election to make that determination.”

He received an ovation from the crowd.

In an appearance on Fox News on Wednesday, the same day as the fund-raising luncheon, Mr. Cruz said he liked and respected Mr. Trump and “I don’t anticipate that changing at all.”

“The reason why I won’t get engaged in personal insults and attacks, I don’t think the American people care about a bunch of politicians bickering like schoolchildren,” said Mr. Cruz, adding at another point, “I’m grateful Donald Trump is running.”

Mr. Cruz has been notable in a field that has grappled with how to handle Mr. Trump for his refusal to publicly criticize his opponent.

Mr. Cruz and his team, who covet Mr. Trump’s supporters, are also keenly aware of the searing criticisms that Mr. Trump has lobbed at rivals who went after him. Those attacks have precipitated a decline in some of their standings in the polls.

Mr. Cruz moved ahead of Mr. Trump in a poll this week of Iowa caucusgoers. But a second poll later the same day showed Mr. Trump leading in the state by a large margin.

Mr. Cruz has in recent weeks nipped at Mr. Trump, but obliquely. He at one point described some rhetoric in the race about immigrants as “unhelpful,” the furthest he has gone in publicly criticizing Mr. Trump, who has drawn withering attacks from Democrats and some Republican leaders for his remarks on Muslims.

Mr. Cruz told reporters on Tuesday that he disagreed with Mr. Trump’s proposed barring of Muslims from entering the country, but went on to praise him for “standing up and focusing America’s attention on the need to secure our borders.”

Mr. Cruz was later asked if he would support Mr. Trump as the Republican nominee.

“I will absolutely support the Republican nominee,” he said, “but I hope and intend for that nominee to be me.”