"Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing," Trump, the Republican presidential nominee, told reporters.
Trump made the remark at a testy news conference at his Doral golf resort in Florida that allowed him to steal some of the limelight from the Philadelphia convention where Clinton on Thursday will accept the Democratic presidential nomination for the Nov. 8 election.
The Clinton campaign shot back that Trump was posing a possible national security threat by encouraging a foreign power to conduct espionage in the United States. Some intelligence experts said the comments raised questions about Trump's judgment.
A spokesman for Trump, Jason Miller, tried to tamp down the storm of protest, saying Trump did not urge Russia to hack Clinton's emails.
Trump said on Twitter that if anyone had Clinton's emails, "perhaps they should share them with the FBI!"
The criticism of Trump's comments reverberated at the Democratic National Convention where speakers brought up the episode to try to intensify Democratic support for Clinton, who is running neck and neck with Trump in the polls, Reuters reported.
"Donald Trump today once again took Russia’s side. He asked the Russians to interfere in American politics," longtime Clinton supporter and former CIA Director Leon Panetta said. "Donald Trump ... is asking one of our adversaries to engage in hacking or intelligence efforts against the United States of America to affect the election."
Another speaker, retired US Rear Admiral John Hutson, said of Trump, "This morning, he personally invited Russia to hack us. That's not law and order, that's criminal intent."
Trump was referring in his remarks to a private email system Clinton kept while secretary of state from 2009 to 2013. She handed over thousands of emails in 2015 to US officials probing that system, but did not release about 30,000 emails she said were personal and not work-related.
Trump dismissed suggestions that WikiLeaks' release of embarrassing Democratic Party emails last week was engineered by Russia to help nudge the US election toward Trump.
"It's just a total deflection, this whole thing with Russia," said Trump. "I never met Putin. I don't know who Putin is."
The Democratic Party chairwoman, Debbie Wasserman Schultz, resigned on Sunday after the leaked emails showed party leaders favoring Clinton over her rival, US Senator Bernie Sanders, for the presidential nomination.