By Emily Stephenson and Alana Wise

WASHINGTON (Reuters) - Republican presidential nominee Donald Trump dug deeper in his efforts to cast doubt on the legitimacy of the U.S. election, saying on Twitter on Sunday that he believed the results were being “rigged” at many polling places.

His tweet came hours after his vice presidential running mate, Mike Pence, said Republicans would accept the outcome of the Nov. 8 contest between Trump and his Democratic rival, Hillary Clinton.

“The election is absolutely being rigged by the dishonest and distorted media pushing Crooked Hillary - but also at many polling places - SAD,” Trump wrote on Twitter, in the latest of a series of comments he has made over the past several days calling into question the fairness of the election.

Trump, who is trailing Clinton in opinion polls, did not provide any evidence to back his allegations of impropriety at the voting booth. Early voting and voting by mail have begun in many states.

In another tweet later on Sunday, Trump blamed “animals representing Hillary Clinton” and Democrats in North Carolina for an overnight attack on a local Republican Party headquarters in that state.

Local authorities said the building in Hillsborough, North Carolina, had been struck through a front window with flammable material and an adjacent building wall was spray-painted with a swastika and the words “Nazi Republicans leave town or else.”

“Animals representing Hillary Clinton and Dems in North Carolina just firebombed our office in Orange County because we are winning,” Trump tweeted.

Clinton denounced the attack as “horrific and unacceptable” in her own tweet, adding: “Very grateful that everyone is safe.

‘COORDINATED EFFORT’

Trump, a New York businessman, who has never held elective office, has often said the electoral process is skewed against him, including during the Republican nominating contests, when he disputed the method for winning delegates to the Republican National Convention.

His latest complaint of media bias stems from allegations by women that he groped them or made other unwanted sexual advances, after a 2005 video became public in which Trump was recorded bragging about such behavior. He apologized for the video but has denied each of the accusations.

“Election is being rigged by the media, in a coordinated effort with the Clinton campaign, by putting stories that never happened into news!” Trump tweeted on Sunday, a sentiment he also expressed in posts and during rallies in Maine and New Hampshire on Saturday. The comments raised questions both from Republicans and Democrats about whether he would accept the outcome should he lose to Clinton.

Trump said after the first presidential debate in September that he would “absolutely” accept the election outcome. But a few days afterward, he told the New York Times: “We’re going to see what happens.”

He has also urged his supporters to keep an eye on voting locations to prevent a “stolen” election, which some critics interpreted as encouraging them to intimidate voters.

Pence said on Sunday he and Trump would respect the will of the voters.

“We’ll respect the outcome of this election,” said Pence, the Indiana governor. “Donald Trump said in the first debate that we’ll respect the will of the American people in this election. The peaceful transfer of power is a hallmark of American history.”

In a weekend statement quoted by media, a spokeswoman for U.S. House of Representatives Speaker Paul Ryan, the top U.S. elected Republican, said: “Our democracy relies on confidence in election results, and the speaker is fully confident the states will carry out this election with integrity.”

‘CONSEQUENCES TO RUSSIA’

On Sunday, Pence for the second time in recent weeks broke from Trump on Russia, this time on that country’s possible involvement in email hacks tied to the U.S. election, saying Moscow should face “severe consequences” if it has compromised U.S. email security.

“I think there’s no question that the evidence continues to point in that direction,” Pence said. “There should be severe consequences to Russia or any sovereign nation that is compromising the privacy or the security of the United States of America,” Pence said on “Fox News Sunday.”

Russian President Vladimir Putin said last week his country was not involved in trying to influence the U.S. election.

Trump, who has been criticized for appearing too close to Russia after he publicly praised Putin’s governing style, has questioned the reports of Moscow’s involvement. “Maybe there is no hacking,” he said during last week’s second debate with Clinton.

Trump also contradicted Pence on Russia during that debate. Pence had said the United States should use military force in Syria if Russia continued air strikes to prop up President Bashar al-Assad, but Trump said he disagreed.

The third and final debate between Trump and Clinton will be on Wednesday in Las Vegas.

(Reporting by Emily Stephenson and Alana Wise; Editing by Bill Trott and Peter Cooney)