Hillary Clinton’s former campaign chairman has told Donald Trump to “get a grip” after the US president attacked him from the G20 summit.
Mr Trump tweeted that “everyone” at the meeting of world leaders in Germany was talking about John Podesta’s supposed role in a Democratic Party email hack.
He accused Mr Podesta of failing to co-operate with US authorities as they investigated the cyber-attack.
Mr Podesta hit back in seven tweets, branding Mr Trump a “whack job”.
“Everyone here is talking about why John Podesta refused to give the DNC server to the FBI and the CIA. Disgraceful!” Mr Trump tweeted on Friday morning from Hamburg.
Mr Trump’s tweet appeared to conflate the hacking of Mr Podesta’s personal account with another cyber-attack.
Mr Podesta pointed out in one of his replies that he did not run the Democratic National Committee (DNC), which suffered a separate email breach.
Russia has been blamed for both leaks but denies the claims.
Mr Trump’s post came shortly before his first face-to-face meeting with Russian President Vladimir Putin on the sidelines of the two-day G20 meeting in Hamburg.
Analysis by Anthony Zurcher, BBC News
Thanks to social media, we once again are offered a look inside Donald Trump’s head at a key point in his presidency.
On the morning of his first face-to-face meeting with Russia President Vladimir Putin, in the midst of a flurry of talks with other world leaders, the president had two things on his mind – former Hillary Clinton campaign head John Podesta and the “fake news media”.
The president once again questioned why the Democratic National Committee – which Mr Podesta is not a part of – did not turn its hacked email server over to government investigators.
DNC says they were never asked by the FBI for access; the FBI insists a request was made. So far, the question has never been resolved definitively – but the FBI, CIA and National Security Agency still say that there is conclusive evidence that Russia was behind the hacks.
Mr Trump and Mr Putin ended their 90-minute meeting on Friday afternoon and, according to the Russian government, cyber-security was one of the topics discussed.
No word yet on whether Mr Podesta’s name came up here, as well. If it did, Mr Trump will surely tweet about it.
Mr Podesta tweeted later on Friday that he was on a cross-country trip when he stopped in West Virginia to learn that the US president had tweeted about him.
“Get a grip man, the Russians committed a crime when they stole my emails to help get you elected President,” Mr Podesta wrote.
“Maybe you might try to find a way to mention that to President Putin.”
Mr Podesta added: “Dude, get your head in the game. You’re representing the US at the G20.”
Adam Schiff, the top Democrat on the House Intelligence Committee, which is investigating alleged Russian meddling in the US election and whether Trump campaign officials colluded, aimed a sarcastic tweet at the president.
“Yes, I’m sure that’s the big talk at G20,” he posted. “Not climate change or trade, but why didn’t John Podesta give a server that wasn’t his to the CIA.”
Several American intelligence agencies have concluded Russia meddled in last year’s US elections, but Mr Trump has as recently as this week said he thinks other countries could have been responsible.
The Washington Times reported this week that federal investigators, who are looking into Moscow’s alleged attempt to sway the US election, have been unable to obtain the hacked computer server from the DNC.
The leaked DNC emails revealed that party officials favoured Mrs Clinton in the tussle for the presidential nomination, a revelation that soured supporters of her liberal challenger, Bernie Sanders.