Republican presidential hopeful Ted Cruz won by wide margins over front-runner Donald Trump in party caucuses in the US states of Kansas and Maine Saturday, but Trump prevailed in two other states.

The latest results in the state-by-state presidential nominating process failed to bring resolution to the race that has seen Trump's upstart campaign gather a strong following even as the party establishment has sought to stop his rise.

Meanwhile, Democratic dark horse Bernie Sanders defeated Hillary Rodham Clinton in the Midwestern states of Kansas and Nebraska.

Saturday's primary elections in five states – four for Republicans and three for Democrats - were the first since Super Tuesday, in which a dozen states voted and solidified the front-runner status of both Trump and Clinton to head the two main parties for the November election.

Trump beat Cruz in the southern state of Louisiana with 41 per cent to 38, and in Kentucky with 36 per cent to 32.

Clinton also won the Democratic contest in Louisiana by 71 per cent to Sanders' 23 per cent.

Both Trump and Cruz called for the field of four candidates to narrow and allow them to face off against each other.

"As long as the field remains divided, it gives Donald an advantage," Cruz said.

Trump called specifically on trailing rival Marco Rubio to drop out.

"Marco Rubio had a very bad night and personally I'd call on him to drop out of the race," Trump said. "I think it's time. You’ve got to be able to win."

“I’d love to take on Ted one on one," he said, claiming that Cruz would not be able to win primary contests in large states like New York and California. Cruz meanwhile maintains that the party would unite with him against Trump.

In Kansas, Cruz won 48 per cent of ballots to Trump's 23 per cent, party officials said. Rubio had 17 per cent and Ohio Governor John Kasich had 11 per cent.

Both Rubio and Kasich are counting on wins in their home states of Florida and Ohio later this month to breathe life back into their campaigns. Both are key electoral states in US elections.

In Maine, Cruz won 46 per cent to Trump’s 33 per cent. Kasich was third with 12 per cent and Rubio took only 8 per cent, party officials said.

The Republican establishment has been struggling to prevent outsider Trump from seizing the conservative party's nomination, and Cruz’s victories Saturday helped to bolster his claims as the conservative alternative.

Cruz also received a psychological boost from a survey of conservative activists meeting outside Washington who named him their choice for president.

Poor showings for Rubio damaged his claim as the establishment favourite, despite endorsements by many prominent party lawmakers.

Trump has remained well ahead in Republican polls, though he has not yet amassed enough delegates to put him entirely out of reach of his competitors.

This week both Mitt Romney and John McCain, the 2012 and 2008 Republican presidential nominees respectively, spoke forcefully against Trump being the nominee.

In the Democratic races, Clinton leads voter surveys over Sanders, and also leads in the tally of delegates needed to secure the nomination during a party convention in July.

Sanders won 68 per cent of the vote in Kansas caucuses to Clinton’s 32 per cent. In neighbouring Nebraska, he won 55 per cent to Clinton’s 45 per cent.

However in Louisiana, where there are more minority voters loyal to Clinton, she secured 70 per cent of the vote.

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