Trump made the comment during a press conference at the Trump National Doral golf resort in Florida.
"Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing," Trump said, referring to emails Clinton said she deleted from her private server because she said they were personal in nature and not tied to her job as secretary of state at the time.
"I think you will probably be rewarded mightily by our press," Trump added.
The Clinton campaign quickly issued a statement condemning Trump for creating a "national security issue."
"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," said Jake Sullivan, Clinton's senior policy adviser. "That's not hyperbole, those are just the facts. This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue."
Trump spent nearly an hour answering questions from the media. When asked about having ties to Russia, Trump said the accusation is "just a total deflection, this whole thing with Russia."
"By the way they hacked, they probably have her 33,000 emails," Trump said, speaking of Russia. "I think they do. They probably have her 33,000 emails that she lost and deleted because you'd see some beauties there. So let's see."
Trump would later reinforce his comment through a message on Twitter.
"If Russia or any other country or person has Hillary Clinton's 33,000 illegally deleted emails, perhaps they should share them with the FBI!" Trump tweeted.
Meanwhile, when Trump was asked if he was behind the hack of the Democratic National Committee's emails last week -- unrelated to Clinton's email server -- he denied the allegation but said he would desire having the ability.
"Honestly it's so far-fetched. It's so ridiculous. Honestly I wish I had that power. I'd love to have that power," Trump said.
U.S. intelligence agencies have blamed the Russian government for the DNC hack, The New York Times reported Tuesday, though they said it's unclear if it was part of routine espionage or an attempt to harm Clinton's campaign. Julian Assange, the founder of Wikileaks, which published the emails Friday, said he hoped the DNC emails would negatively affect Clinton.