The push to stop Trump comes as the billionaire real estate tycoon finds himself at the centre of a firestorm over his call to ban Muslims from entering the United States.
Donald Trump calls for barring Muslims from entering US
The Washington Post reported on Thursday that Republican powerbrokers argued at a recent dinner that the party should plan to coalesce around an alternative at the nominating convention if Trump storms through the primaries.
"I think they're making a big mistake. I think I'm the one to beat Hillary," Trump told Fox News Sunday, referring to the Democratic frontrunner Hillary Clinton.
"I think they are rather kidding themselves."
A new NBC/Wall Street Journal poll shows Trump in the lead with 27 percent among Republicans nationally, and other polls give him similar leads in many state contests for the party's presidential nomination.
But the poll also showed that Cruz, a conservative senator who has been careful not to publicly criticise Trump, has surged into second place nationally with 22 percent.
And a Des Moines Register poll out Sunday said Cruz was leading by 10 points among Republicans in Iowa, site of the first nomination contest February 1.
"It's Cruz-mentum. Ted Cruz is crushing it in Iowa," Jennifer Jackson of the Des Moines Register said in a video on the paper's website.
Cruz's 31 percent showing in Iowa versus Trump's 21 percent followed a December 7 Monmouth University poll that also put the Texas senator in the lead in the state.
Cruz has surged as retired neurosurgeon Ben Carson has faded in Iowa, a rise and fall attributed to renewed focus on security issues in the wake of the December 2 attack that left 14 dead in California.
Trump ban on Muslims 'endangers national security': Kerry
Carson polled fourth in the NBC/Wall Street Journal poll at 11 percent, and third in Iowa at 13 percent in the Des Moines Register poll. Meanwhile Florida Senator Marco Rubio was third in the national poll with 15 percent and fourth in Iowa with 10 percent.
Trump's response to the Cruz challenge was that the Texas senator lacked the temperament and judgment to be president.
"The way he's dealt with the Senate, where he goes in frankly like a bit of a maniac. You never get things done that way," he said.
"You can't walk into the Senate and scream and call people liars and not be able to cajole and get along with people. He'll never get anything done. That's the problem with Ted," he said.
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