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.