Donald Trump told a Florida news conference on Wednesday that Russia ought to seek out "30,000 emails that are missing" from his rival Hillary Clinton's account, from when she was secretary of state.
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"Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing," Trump said.
Politico reports Trump telling Moscow "I think you will probably be rewarded mightily by our press" were Russian authorities to come up with the electronic messages in question.
Later on, when asked if he was really urging a foreign country to hack into Clinton’s private email server to meddle in the presidential election, Trump told the reporter to “be quiet.” Adding, “That’s up to the president. Let the president talk to them.”
Separately, the Republican presidential nominee dismissed any suggestion that Russia was behind the theft and release of embarrassing Democratic Party emails made public last week.
"It is so far-fetched, it's so ridiculous," Trump said, suggesting that China or some other party could be involved.
Trump also sought to minimize his touted relationship with Russian leader Vladimir Putin.
"I don't know who Putin is. He said one nice thing about me I never met Putin," Trump said. "I have nothing to do with Russia. I said that Putin has much better leadership qualities than Obama, but who doesn't know that?"
Clinton, due on Thursday to accept the Democratic Party nomination to face Trump in the Nov. 8 election, kept a private system for her emails at her New York home while serving as secretary of state from 2009-2013.
Those emails were the subject of a Federal Bureau of Investigation probe that found no basis for criminal charges despite what FBI Director James Comey called evidence Clinton was "extremely careless" in her handling of classified information.
Putin's name came up this week after the exposure by Wikileaks of embarrassing emails of leading Democrats, which forced the chairwoman of the Democratic National Committee, Debbie Wasserman Schultz to resign just ahead of the party convention.
Clinton's campaign denounced Trump's comments, with Jake Sullivan, a senior policy adviser to the Clinton campaign, saying that "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," Sullivan said in a statement. "This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue."
On Wednesday, U.S. President Barack Obama said it was possible Russia would try to influence the U.S. presidential election after a leak of Democratic National Committee emails that experts have blamed on Russian hackers. The Kremlin dismissed the allegations as "horror stories."
Trump's running mate, Mike Pence, took a different approach from Trump to the release of the Democratic Party emails, saying the FBI would get to the bottom of the matter.
"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," Pence said in a statement.
Reuters contributed to this report