Former US president Donald Trump won the key New Hampshire primary on Tuesday, moving him ever closer to locking in the Republican presidential nomination and securing a White House rematch with President Joe Biden.

Trump, 77, attacked his sole remaining challenger Nikki Haley in a rambling victory speech and said that when the primary contest reaches her home state of South Carolina, "we're going to win easily".

In her speech, Haley insisted the race was "far from over" and told supporters that Democrats actually want to run against her former boss, due to his record of sowing "chaos".

"They know Trump is the only Republican in the country who Joe Biden can defeat," Haley, 52, said.

Haley had hoped for a major upset in the northeastern state, but Trump — her former boss when she was UN ambassador during his administration — won by around 54 percent to 43 percent, with more than 90 percent of votes counted.

At one point swearing on prime-time TV, Trump said the United States was a "failing country" and claimed that undocumented migrants were coming from psychiatric hospitals and prisons and "killing our country".

"It is now clear that Donald Trump will be the Republican nominee," Biden responded by saying in a statement.

"I think it's a two-person race now between Trump and Biden," Keith Nahigian, a veteran of six presidential campaigns and former member of Trump's transition team, told Agence France-Presse.

New Hampshire was markedly more Haley-friendly than the states she will subsequently face, should she stay in the race, South Carolina will be a tough sell, the agency reported.

Trump won a crushing victory in the first Republican contest in Iowa last week, with Haley a distant third.

What was once a crowded field of 14 candidates then narrowed to a one-on-one matchup on Sunday after Florida Governor Ron DeSantis dropped out. No Republican who won both Iowa and New Hampshire has ever lost the nomination battle, according to The Associated Press.

The race now turns to Nevada, where Trump is already claiming an all-but-certain victory, and next month to Haley's home state of South Carolina, where he leads the former governor by some 30 percentage points.

Trump was already the leader in national Republican polling, despite piles of legal challenges as he faces 91 criminal charges related to everything from seeking to overturn the 2020 presidential election to mishandling classified documents and arranging payoffs to an adult movie actress, AP reported.

Trump's trial in the defamation case brought by the writer E. Jean Carroll is scheduled to resume on Thursday, and will not be held on Wednesday as initially scheduled, Manhattan federal court records showed on Tuesday.

On the Democratic side, Biden won the unofficial Democratic primary in New Hampshire, giving him a symbolic boost, AFP reported.

The Democratic National Committee voted to start its primary next month in South Carolina, but New Hampshire pushed ahead with its own contest.

Biden didn't campaign or appear on the ballot but topped a series of little-known challengers. The president marked the day by campaigning alongside Vice-President Kamala Harris in Virginia at a rally for abortion rights.

With Trump touting his role in the ending of the constitutional right to abortion, Biden told an enthusiastic crowd that the Republican was "hellbent" on further restrictions.

Biden faces his own challenges. There are widespread concerns about his age at 81 years old. Dissent is also building within his party over Biden's alliance with the ongoing conflict in Gaza, putting the president's standing at risk in swing states like Michigan.

Meanwhile, AP reported that Canada's government is preparing for the possibility that Trump could reach the White House again and the "uncertainty" that would bring, Prime Minister Justin Trudeau said on Tuesday.