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Donald J. Trump’s support among Republicans has cracked the 40 percent threshold, according to a national poll released on Monday that shows the billionaire real estate mogul pulling away from his rivals after his proposal to bar Muslim immigrants from entering the United States.

The Monmouth University survey showed Mr. Trump’s backing at 41 percent, up from 28 percent in a poll conducted by the same group in October. Senator Ted Cruz (14 percent) of Texas, Senator Marco Rubio (10 percent) of Florida and Ben Carson (9 percent) are competing for second place. The poll came as another survey of Iowa Republicans showed Mr. Trump deadlocked with Mr. Cruz.

Patrick Murray, director of the Monmouth University Polling Institute, said that Mr. Trump’s gains nationally underscore his ability to please his supporters while turning off the Republican establishment.

“It has become abundantly clear that Trump is giving his supporters exactly what they want, even if what he says causes the G.O.P. leadership and many Republican voters to cringe,” Mr. Murray said.

Mr. Trump has seen an upward bounce in his national poll numbers after his proposal this month about foreign Muslims. Polls show Republicans are split on the idea, while the country over all is largely against it.

Although Mr. Trump is showing no signs of fading nationally, he is under increasing pressure from Mr. Cruz in Iowa, where many evangelical voters have shifted their support to the senator from Texas instead of Mr. Carson.

The Monmouth poll found that two-thirds of Republican voters think Mr. Trump has the right temperament to be president. He is showing that his support is broad-based within the party, and he is drawing particularly well with voters who associate themselves with the Tea Party and those with lower education levels.

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

A separate poll in Iowa that was released Monday was the third survey in the state to confirm Mr. Cruz’s momentum there. But the poll painted a less dire picture for Mr. Trump than the earlier surveys among likely Republican caucusgoers.

Mr. Trump led among likely Iowa caucusgoers with 28 percent, statistically tied with Mr. Cruz at 27 in the poll from Quinnipiac University. The gap in Iowa is similar to a two-percentage point lead Mr. Trump held over Mr. Cruz in the same pollster’s survey three weeks ago.

It followed two weekend polls showing a big surge for Mr. Cruz: one by The Des Moines Register giving the Texas senator a 10-point lead, and another by Fox News showing Mr. Cruz ahead, 28-26 percent.

Mr. Cruz’s strength with conservatives was confirmed in the Quinnipiac Iowa poll. He led Mr. Trump among groups of voters who described themselves as evangelicals, Tea Party supporters and very conservative.

Mr. Trump led with somewhat and moderately conservative voters, and those without a college degree.

With the terrorist attacks in Paris and San Bernardino, Calif., reshaping the Republican race, voters were split about which of two front-runners was best prepared: Likely Republican caucusgoers said Mr. Trump would best handle terrorism and Mr. Cruz was best on foreign policy. The interviews were conducted Dec. 4-13; Mr. Trump’s statement on barring Muslims was issued on Dec. 7.

The two men were far ahead of their nearest challengers, with Mr. Rubio getting 14 percent and Mr. Carson at 10 percent.

The poll showed Mr. Trump was viewed less favorably than all three of the other top four contenders.

“One good sign for Senator Cruz in his battle with Trump is that voters view him much more favorably than Trump, meaning that the Texas lawmaker may have a higher potential upside,’’ Peter A. Brown, assistant director of the Quinnipiac poll, said in a statement.

With the Iowa caucuses 49 days away, the horse race has the potential fur further shake-ups before voting on Feb. 1. In past cycles, many voters made up their minds the last several days, according to entrance polls. That is good news for candidates now in single digits: former Gov. Jeb Bush of Florida at 5 percent, Senator Rand Paul of Kentucky at 4 percent, and Gov. Chris Christie of New Jersey and Carly Fiorina at 3 percent.