WASHINGTON: Billionaire Donald Trump notched a pair of easy victories Tuesday in the Republican presidential nomination race, as Democrat Hillary Clinton suffered a surprise loss to Bernie Sanders in one of the two major states at stake.

Trump shrugged off a barrage of negative advertising and intense efforts by the party establishment to derail his White House campaign to win primaries in Michigan and Mississippi, signaling to his rivals he can survive anything they throw at him.

"I don't think I've had so many horrible, horrible things said about me in one week, $38 million worth of horrible lies," Trump told a crowd in Florida as he celebrated his victories. "I think we ought to use that money to fight Hillary Clinton and the Democrats."

It was not a clean sweep for Trump, however. He came second to Senator Ted Cruz, his nearest Republican competitor, in Idaho. A fourth Republican vote was taking place in Hawaii, with results expected later.

Clinton, the former secretary of state, handily defeated Sanders in the southern Gulf state of Mississippi, thanks to a strong turnout by African-Americans who comprised a majority of Democratic voters there. But she slumped to defeat in the industrial rust belt state of Michigan, where Sanders was ahead by two percentage points with 93 per cent of precincts reporting.

"This has been a fantastic night in Michigan," Sanders said shortly before the race was called in his favour.

Despite the upset, Clinton received a psychological boost by passing the halfway point in the all-important race to reach the 2,382 delegates needed to win the Democratic nomination.

But even with the delegate math in Clinton's favor, Sanders's strong showing will re-energise his campaign, and raise questions about her ability to win key industrial states in the general election, such as Michigan, Ohio and Pennsylvania.

Sanders, a US senator from Vermont and self-described democratic socialist, has energised young voters with calls for greater economic equality and denunciations of what he sees as a corrupt US political system.

Poor Rubio showing

Clinton has now won 13 out of 22 nomination contests, with Trump prevailing in 14 out of 23 races as the two inch closer to the tipping point in their respective races.

Rubio, the senator whom mainstream Republicans rallied behind as the man to topple Trump, suffered another poor showing, facing the prospect of receiving zero delegates from either Michigan or Mississippi, the two main prizes of the night.

Trump has already called on Rubio to drop out of the race, but the senator has vowed to stay in at least until his home state of Florida ─ which with 99 delegates at stake is a major prize on the primary calendar ─ votes on March 15.

"They didn't do so well tonight, folks," Trump said in a victory speech in Florida, referring to his Republican rivals. "Only one person did well tonight: Donald Trump."

The braggadocious Trump's caustic style has angered some voters ─ and fellow Republicans ─ but he insisted Tuesday he could draw millions more to his movement.

"We'll take many, many people away from the Democrats," he said. "We're seeing that. We had people come over here who have never voted Republican."

In a bizarre scene, Trump spent several minutes hawking some of his companies' odder products ─ water, steaks, wine, Trump vodka, even his Trump University ─ which establishment critics had berated as examples of his failed enterprises.

With his latest big wins ─ claiming 47.3 per cent in Mississippi and 36.5pc in Michigan based on near-final results ─ Trump solidifies his claim to have the broadest appeal among the Republican electorate as he marches toward the nomination.

But a new Washington Post poll of Republican-leaning registered voters shows Trump with 34pc support nationwide, compared with 25pc for Cruz, 18pc for Rubio and 13pc for Kasich.

That is a tighter race than in January, when the Post showed Trump up 16 points against Cruz and 26 against Rubio.

Cruz, the 45-year-old champion of the religious right, has done well in delegate-rich Texas and nearby states and is nipping at the billionaire real estate mogul's heels.

A total of 150 Republican delegates were up for grabs Tuesday out of 1,237 needed to win the party's nomination.

As of Monday Trump has 384 delegates, compared with 300 for Cruz, 151 for Rubio and 37 for Kasich. Trump is expected to win dozens more after Tuesday.

Clinton was hoping Tuesday could be the day she pivots toward the GOP.

"Every time you think it can't get any uglier, they find a way," she told supporters at a rally in Ohio.

"And as the rhetoric keeps sinking lower, the stakes in this election keep rising higher. Now, running for president shouldn't be about delivering insults. It should be about delivering results for the American people."