US Republican presidential candidate Donald Trump has urged Russia to find the missing emails of Hillary Clinton, while she was secretary of state, drawing condemnation from his Democratic rival.

"Russia, if you're listening, I hope you're able to find the 30,000 emails that are missing," Trump told reporters in Miami on Wednesday. "I think you'll probably be rewarded mightily by our press!"

Trump was referring to emails on Clinton's private email server that she deleted because she said they were private before she turned other messages over to the State Department. The FBI declined to prosecute Clinton over her email practices but its director said she had been "extremely careless" handling classified materials.

Leaked emails appear to show DNC hostility to Sanders

Clinton's campaign quickly responded to Trump's latest statement, calling it the "first time that a major presidential candidate has actively encouraged a foreign power to conduct espionage against a political opponent.

"This has gone from being a matter of curiosity, and a matter of politics, to being a national security issue," Clinton's campaign said in a statement.

But shortly after Trump's comments, his running mate, Indiana Governor Mike Pence, took a different stance and warned of "serious consequences" if Russia interfered in the election.

"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.

Russia has brushed aside suggestions it was involved. "I don't want to use four-letter words," Russian Foreign Minister Sergei Lavrov told reporters on Tuesday.

WikiLeaks emails

In the same news conference on Wednesday, Trump dismissed suggestions that Russia had sought to influence the US election by engineering the theft of Democratic Party emails released by WikiLeaks last week.

The Democratic Party chairwoman, Debbie Wasserman Schultz, resigned on Sunday after the leaked emails showed party leaders favouring Clinton over her rival, Senator Bernie Sanders, for the presidential nomination.

Al Jazeera's Kimberly Halkett, reporting from Philadelphia, where the Democrats are holding their presidential convention, said Trump's latest statement could help Clinton's party deflect the criticism and internal dissent they had been facing following the leaks.

"This seems to be a welcome gift for the Democratic Party, given the fact that they are trying to distance themselves from the content, which shows that they colluded in favour of Hillary Clinton's candidacy" over Sanders, Halkett said.

Representatives from Clinton's campaign have previously claimed that Russians hacked computers belonging to their party, and released those emails on the eve of the party's convention to benefit Trump's candidacy.

'Serious consequences'

But Trump dismissed the claims of his role in the leak, saying it was not clear who hacked those emails, and describing the incident a sign that foreign countries no longer respected the US.

Trump, whom Democrats have accused of having cozy ties with Russian President Vladimir Putin, repeatedly declined to condemn the actions of Russia or any other foreign power of trying to intervene in the a US election.

He also has downplayed his affection for Putin and said he would treat the Russian leader "firmly", though he said he wanted to improve relations with Moscow.

Elie Jacobs, a cyber-security expert from the Truman National Security Project, said it was likely that Russia was involved with the hacking - but doubted links to Trump.

"I think this was purely an act of cyber warfare, and that's really the larger story, and less about what it has to do with the political cycle right now," Jacobs told Al Jazeera.

US President Barack Obama, meanwhile, said "anything's possible" when asked during an interview whether the Russians could be working to sway the election toward Trump.

"Donald Trump has repeatedly expressed admiration for Vladimir Putin," Obama said during the sit-down with NBC News that aired on Tuesday.

"And I think that Trump's gotten pretty favourable coverage back in Russia."