Republican presidential hopeful Ted Cruz split victories in four nominating contests with front-runner Donald Trump on Saturday, bolstering Cruz's argument that he represents the party's best chance to stop the brash New York billionaire.

The results were a repudiation of a Republican establishment that has bristled at the prospect of either Cruz or Trump winning the party's nomination and has largely lined up behind US Senator Marco Rubio of Florida, who was shut out in all four contests.

"I think it's time that he dropped out of the race," Trump said of Rubio after the contests. "I want Ted one on one."

Cruz won Kansas and Maine on Saturday, while Trump won the bigger states of Louisiana and Kentucky, holding onto his lead in the race for the Republican nomination for the November 8 presidential election, even though Cruz captured more delegates on Saturday.

The next big contest, and a crucial one, will be Tuesday's primary in the industrial state of Michigan. Republicans in three other states, Mississippi, Idaho and Hawaii, also will vote on Tuesday. Puerto Rico Republicans will vote on Sunday.

In the Democratic race, front-runner Hillary Clinton won in Louisiana, and her rival Bernie Sanders, a US senator from Vermont, won in Kansas and Nebraska, in results that did not substantially change Clinton's big delegate lead.

Mainstream Republicans have blanched at Trump's calls to build a wall on the border with Mexico, round up and deport 11 million undocumented immigrants and temporarily bar all Muslims from entering the US.

But the party's establishment has not been much happier with Cruz, who has alienated many party leaders in Washington.

"It looks like it will be the angry Trump voters against the purist conservative Cruz voters," said Washington-based Republican strategist Ron Bonjean. "The establishment is just being left out."

A spokesman for Rubio, who spent the past week launching harsh personal attacks on Trump, said the senator would push on with an eye on the March 15 contest in Florida.

"After we win the Florida primary, the map, the momentum and the money is going to be on our side," spokesman Alex Conant said in a statement.

Cruz, a first-term US senator from Texas who has promoted himself as more of a true conservative than Trump, said the results showed he was gaining momentum in the race to catch the real estate mogul.

Cruz, 45, has run as an outsider bent on shaking up the Republican establishment in Washington. A favourite of evangelicals, he has called for the US to "carpet bomb" the Islamic State militant group and has pledged to eliminate the tax-collecting Internal Revenue Service and four cabinet agencies and to enact a balanced budget amendment. - Reuters