Republican presidential front-runner Donald Trump had a new target on Sunday, calling fellow White House contender Ted Cruz "a little bit of a maniac" as the U.S. senator surpassed him in an Iowa poll.

Cruz's dogged pursuit of conservative Iowa voters has paid off in the form of a 10-point lead over Trump in the state, which has one of the earliest presidential contests. Unlike the other Republicans in the 2016 White House race, the U.S. senator from Texas has embraced Trump and avoided public criticism of the popular candidate.

But last week, he questioned Trump's judgment at a private fundraiser, according to the New York Times, after the billionaire businessman advocated temporarily banning Muslims from entering the United States. That got Trump's attention.

"I don't think he is qualified to be president," Trump said on Fox News Sunday. "I don't think he has the right temperament. I don't think he's got the right judgment. When you look at the way he has dealt with the Senate, where he goes in there like a, you know, frankly, like a little bit of a maniac — you are never going to get things done that way."

Trump touted his ability to get along with liberals and conservatives and said that was the hallmark of the "world-class businessman" he is.

'He's been so nice to me'

The front-runner, whose comments on Muslims drew widespread criticism but may not dent his lead in several national public opinion polls, made a sarcastic reference to Cruz's respectful treatment of him.

"He's been so nice to me. I mean I could be saying anything and he'd say, I agree I agree," Trump said on CNN's State of the Union.

On the Fox program, he also criticized Cruz for talking about him behind his back. Cruz rose to 31 per cent, above Trump's 21 per cent, in an Iowa poll released on Saturday by the Des Moines Register and Bloomberg News. That's a 21-point jump from October.

From left to right, Republican U.S. presidential candidates Donald Trump, Ben Carson and Ted Cruz appear at the Republican presidential debate on Nov. 10. Cruz surpassed Trump in recent Iowa poll at the expense of Carson, who dropped to third. (Jim Young/Reuters)

His rise came at the expense of retired neurosurgeon Ben Carson, who dropped to third with 13 per cent in the poll, while U.S. Senator Marco Rubio hovered at 10 per cent.

Jeb Bush was at 6 per cent, a 1 per cent increase from October. Rubio, who has seen an uptick in his own poll standings in recent weeks, criticized his Senate colleague on defence spending, saying Cruz talked about carpet-bombing Islamic State while voting to cut the military budget.

Rubio was measured in his criticism of Trump on NBC's Meet the Press, saying, "There's a lot we have a difference of opinion on, but we can't ignore that he's touched on some issues that people are concerned about."

Trump made the same point on Sunday, saying Americans are living in fear of being attacked. He linked his proposal to ban Muslims temporarily to his tough-on-immigration ideas, which included building a wall at the border with Mexico.

"One of the reasons I'm sitting here and one of the reasons I'm so high in the polls is because it all started with the borders," Trump told CNN.

"I took much more heat when I said illegal immigration and southern borders and the wall and all of that than I ever took for this."