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Donald J. Trump’s popularity has climbed to its highest level to date less than a month before the first Republican votes will be cast, according to a national NBC/Wall Street Journal poll released Thursday that shows Republicans becoming increasingly comfortable with the idea of the billionaire businessman being their presidential nominee.

The survey found that 33 percent of likely Republican primary voters support Mr. Trump, with Senator Ted Cruz of Texas trailing him in second place at 20 percent. Mr. Trump’s lead has more than doubled from a month ago, when he was beating Mr. Cruz by 27 percent to 22 percent.

In recent weeks, Mr. Trump has been attacking Mr. Cruz more directly, raising questions about the senator’s Cuban heritage and his Canadian roots to create doubt over whether he is even eligible to run for president. Mr. Cruz was born in Calgary, Alberta, and his mother is an American citizen.

Trailing Mr. Trump and Mr. Cruz are Senator Marco Rubio of Florida, at 13 percent, and Ben Carson,at 12 percent. Gov. Chris Christie of New Jersey and Jeb Bush are further behind, each with 5 percent.

The crowded field appears to be helping Mr. Trump, according to the poll, which shows him winning a hypothetical three-way race against Mr. Cruz and Mr. Rubio. However, in a head-to-head match-up he trails Mr. Cruz by eight percentage points.

Mr. Trump has also made significant headway when it comes to voters being comfortable with the idea of backing him.

In March, just 23 percent of Republican primary voters said they could envision themselves supporting his candidacy. Now, 65 percent say they could see themselves supporting him. Mr. Bush’s fortunes have faded almost as drastically as Mr. Trump’s have risen, as just 42 percent can see themselves supporting the former Florida governor, down from 75 percent in June.

The poll had a margin of sampling error of plus or minus five percentage points.