Washington (AFP) - Republicans struggled to deal with the fallout from Donald Trump's widely condemned remarks on Muslims Wednesday, worrying the controversial mogul could torpedo their 2016 White House hopes.

The party -- which hopes to end eight years of Democratic White House occupancy -- faces a stark choice between turning on their presidential frontrunner and tethering a 161-year-old brand to the whims of a billionaire many Americans see as a bigot.

Trump caused international outrage Monday, when he demanded a ban on Muslims traveling to the United States.

He currently leads Republican polls by double digit margins over his nearest rivals.

His fellow Republican candidates were among those to condemn that plan, but senior party figures have refused to throw Trump under the bus, or even rule out voting for him.

"I like and respect Donald Trump," said Senator Ted Cruz, who is a favorite of conservative Republicans.

"I continue to like and respect Donald Trump. While other candidates in this race have gone out of their way to throw rocks at him, to insult him, I have consistently declined to do so and I have no intention of changing that now."

But sensing trouble, Trump very bluntly warned Republicans he may launch a third party campaign if they move against him.

That could all but kill Republican chances of beating Hillary Clinton, if she is indeed the eventual Democratic nominee.

"A new poll indicates that 68 percent of my supporters would vote for me if I departed the GOP (Republicans) and ran as an independent," Trump said in a Facebook post Tuesday.

According to a USA Today/Suffolk survey published Tuesday, he is supported by 27 percent of Republican likely voters, a level of support that has been fixed through months of controversy.

His nearest rival, Cruz, stands on 17 percent, with Senator Marco Rubio on 16.

Trump's campaign has thrived on free television and social media exposure garnered by ever-more outlandish remarks.

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Party elders have long taken a long view, stressing he has yet to win any nominating elections, which begin in Iowa in just under two months.

I "still believe that the majority of GOP, which nominated Romney 4 yrs ago, will not nominate Trump," tweeted Katie Packer Gage, a key figure in former governor Mitt Romney's election campaign, which saw him win the nomination but lose to incumbent President Barack Obama.

But Republicans are increasingly wondering whether standing behind Trump could also spell political oblivion.

With the United States demographically shifting to become less white, Republicans can scarcely risk further alienating minority voters, who vote Democratic in droves.

Trump's comments already in the cycle -- and the harsh tone on immigration sounded by others such as Cruz -- may leave the party more dependent than ever on just the white, conservative and evangelical voters who back Trump.

"The Republicans are in a terrible dilemma," said Larry Sabato, director of the Center for Politics at the University of Virginia.

"How are they going to bring the energized anti-establishment Trump wing together with the establishment part of the party?"

"I don't see the obvious way, other than hatred of Hillary Clinton and Barack Obama, that's the only thing that can bring them together.

"It might be enough. But it might not be."