On a day when the election campaign reached new extremes of toxicity and strangeness, Mr Trump held a press conference in Florida where he effectively asked a foreign nation to carry out cyber-spying on his rival for the White House.

“Russia, if you’re listening, I hope you're able to find the 30,000 emails that are missing,” said Mr Trump, speaking at one of his golf resorts. “I think you’ll be rewarded mightily by our press.”

Asked if he was concerned that he was apparently encouraging Russia to spy on an American political party, he added: “It gives me no pause. If Russia or China or any of those country gets those emails, I’ve got to be honest with you, I’d love to see them.”

Mr Trump also appeared to imply that Russian leader Vladimir Putin used the N-word to describe President Barack Obama.

The controversy over the emails follows the publication last week of around 20,000 emails belonging to the Democratic National Committee (DNC) by Wikileaks. The emails revealed a plot to try and smear Ms Clinton’s rival, Bernie Sanders, something that forced the resignation of the DNC chairperson

US media said that intelligence experts believed that the Russian government was behind the hacking of the DNC emails.

Over the weekend, Ms Clinton’s spokesman suggested that Mr Trump, who has previously spoken favourably of Mr Putin, was somehow involved in their release, which came on the eve of the Democratic National Convention in Philadelphia. That claim was denied by Mr Trump.

Donald Trump says Putin called Obama the 'N-Word'

On Wednesday, Mr Trump refused to call on Russia to desist from any such actions. “If it’s any foreign country it shows how little respect they have for the United States,” said Mr Trump, who added that he was “not an email person myself because I believe it can be hacked”.

The comments by the Republican candidate were quickly condemned by the Democrats. Ms Clinton’s senior policy advisor, Jake Sullivan said in a statement: “This has to be the first time that a major presidential candidate has actively encouraged a foreign power to conduct espionage against his political opponent. That's not hyperbole, those are just the facts.”

He added: “This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue.”

Many on social media accused Mr Trump of treason, while CNN political analyst Christine Quinn said: “It’s honestly almost treasonous.

Former CIA Director Leon Panetta told the network that Mr Trump's comments raised questions about his loyalty to the US. “You've got now a presidential candidate who is, in fact, asking the Russians to engage in American politics. I just think that that's beyond the pale. I think that kind of statement only reflects the fact that he truly is not qualified to be president of the United States,” he said.

Mr Trump’s colleagues in the Republican party distanced themselves from his comments. His vice presidential running mate, Governor Mike Pence, said there would be serious consequences if the FBI determined that Russia was behind the recent hacking, or was trying to meddle in the US election.

“If it is Russia and they are interfering in our elections, I can assure you both parties and the United States government will ensure there are serious consequences,” said the Indiana governor.

House Speaker Paul Ryan, a Republican, also put space between himself and his party’s presidential nominee's remarks.

“Russia is a global menace led by a devious thug,” said his spokesman Brenan Buck. “Putin should stay out of this election.”

Russia has denied that it is behind the email hacking or that it is trying to interfere with the US election. It said the allegations were nothing more than ““horror stories”.

“Moscow is at pains to avoid any words that could be interpreted as direct or indirect interference in the election process,” Kremlin spokesman Dmitry Peskov told a conference call, according to Reuters.

“We see that the Russian card is in the red corner on the writing table of all Washington politicians during the election campaign, and that very often they make it a trump card in their game.”

In an interview with NBC, President Obama also said it was possible Russia would try to influence the US presidential election.