US presidential nominee Donald Trump vowed to remain in the race even as his campaign was thrown into crisis following news of a recording of him making lewd comments in 2005.

His running mate Mike Pence criticised him and more than a dozen prominent Republicans withdrew support and urged him to drop out.

Mr Trump told the Wall Street Journal today that there was "zero chance I'll quit".

The video was the latest calamity for Mr Trump, who had hoped to revive his flagging campaign in the face of a recent drop in polls with less than a month until election day.

The 2005 video of Mr Trump talking on an open microphone showed the then-reality TV star speaking openly about groping women and trying to seduce a married woman.

The video was taped only months after Mr Trump married his third wife, Melania.

The backlash against the video was swift and widespread.

In an unusual move, vice presidential running mate Mike Pence issued a critical statement, saying on Twitter that he cannot defend the nominee.

"As a husband and father, I was offended by the words and actions described by Donald Trump in the eleven-year-old video released yesterday," Mr Pence, who is governor of Indiana, said in a statement. "I do not condone his remarks and cannot defend them."

Mr Pence indicated he would continue to support Mr Trump, despite calls from several Republicans that the New York real estate mogul step aside and let Mr Pence be the nominee.

There is no precedent for a major party to replace its nominee this late in the campaign and it is unclear if there is an avenue to force him out.

Voting has begun in several states, including the important swing states of Virginia and North Carolina.

A hastily recorded apology by Trump today did not stymie an avalanche of calls from members of his party to quit.

Mr Trump quickly moved to conduct damage control in a video statement today in which he declared himself a changed man and attempted to shift the focus to his opponent, Democrat Hillary Clinton.

He threatened, again, to focus his attacks on the infidelities of her husband, former president Bill Clinton, saying he would talk more about the pasts of both Clintons with only a month until the 8 November election.

Mr Trump has dismissed questions about his own marital infidelities as irrelevant.

"Anyone who knows me knows these words don't reflect who I am. I said it, I was wrong, and I apologise," Mr Trump said in his video statement, posted on his Facebook page.

The video overshadowed the publication of excerpts of Mrs Clinton's paid closed-door speeches made public yesterday by a hacker who claimed to have obtained them from the email account of John Podesta, the chairman of the Democrat's campaign.

In the speeches, she advocates for more open borders and trade, a position she abandoned during the primary because it was untenable to Democratic progressives. Mr Trump has repeatedly criticised her for her past support of free trade.

Trump has struggled to win over women voters, and the video is expected to further feed Democratic criticism about his past behaviour toward women.

"I did try and f*** her. She was married," Mr Trump said about one woman in the 2005 video, before discussing his attraction to others.

"I just start kissing them," he said. "And when you're a star they let you do it."

"Grab them by the *****. You can do anything," Mr Trump said.