Mrs Clinton did not hand over 30,000 emails as part of an investigation into her private email server as they contained private details.
“Russia, if you’re listening, I hope you’re able to find the 30,000 emails that are missing,” Mr Trump said on Wednesday.
“I think you’ll be rewarded mightily by our press.”
The emails would contain some “beauties”, he said. Soon after, he wrote on Twitter that if anyone had the emails, they should hand them over to the FBI.
His appeal comes as Russia stands accused of hacking emails from the Democratic National Committee (DNC) for Mr Trump’s benefit. Both Russia and Mr Trump deny the allegation.
“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,” Jake Sullivan, Mrs Clinton’s senior policy adviser, said.
“This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue.”
In a statement released within an hour of Mr Trump’s comments, his vice-presidential candidate Mike Pence said there would be “serious consequences” if the FBI could prove Russia was attempting to interfere with the election.