The explosive claim – which could have profound consequences for Mr Trump’s re-election effort if proven true – emerged as Assange, 48, appeared at Westminster Magistrates’ Court ahead of a hearing next week about his possible extradition to the US.
Assange’s barrister highlighted evidence that former US Republican congressman Dana Rohrabacher had been to see Assange in the Ecuadorian embassy in August 2017, in the early days of Robert Mueller’s probe into Russian interference in the previous election.
Edward Fitzgerald QC said a statement from Assange’s lawyer Jennifer Robinson shows “Mr Rohrabacher going to see Mr Assange and saying, on instructions from the president, he was offering a pardon or some other way out, if Mr Assange ... said Russia had nothing to do with the DNC leaks”.
A series of emails embarrassing for the Democrats and the Hillary Clinton presidential campaign were hacked before being published by WikiLeaks in the run-up to the 2016 election.
The Mueller investigation, published in April, found Russian GRU agents hacked Ms Clinton’s private server for the first time just hours after Mr Trump’s public request for Russia to “find the 30,000 emails that are missing”.
While the report found no evidence that Mr Trump's team criminally conspired with Russia, he appeared to have claimed to know about future WikiLeaks drops in advance.
One section of the report, heavily redacted by the justice department, said: “Trump told [former deputy campaign chair Rick Gates] that more releases of damaging information would be coming.”
Mr Gates was cited as saying that after the release of the first batch, “the Trump campaign was planning a press strategy, a communications campaign and messaging based on the possible release of Clinton emails by WikiLeaks”.
It also confirmed Assange and Donald Trump Jr had spoken on Twitter, with the former thanking the president’s son for “talking about our publications”.
But after Assange – who argues his extradition to the US is politically motivated – was arrested in April, the president claimed: “I know nothing about WikiLeaks. It’s not my thing.”
On Wednesday, the White House issued a furious denial of Assange's lawyer's claim that Mr Rohrabacher offered him a quid pro quo at the Ecuadorian embassy on Mr Trump’s behalf.
“The president barely knows Dana Rohrabacher other than he’s an ex-congressman. He’s never spoken to him on this subject or almost any subject,” White House press secretary Stephanie Grisham said in an email.
“It is a complete fabrication and a total lie. This is probably another never ending hoax and total lie from the DNC.”
But several former officials suggested the claims should be taken seriously – just weeks after the president was acquitted of an alleged abuse of power in encouraging Ukraine to investigate his political rival in the upcoming election.
“Lawyer’s don’t usually make claims in court they can’t prove," Joyce White, a former US attorney, tweeted.
“It sure sounds like Assange’s attorneys are prepared to back-up this claim with evidence,” said Ned Price, a former CIA official who served as a special adviser to Barack Obama.
“It’s another indication that Trump’s assault on the rule of law isn’t new; it’s been ongoing throughout his term. And imagine just how much we don’t yet know.”
District Judge Vanessa Baraitser said Assange’s evidence would be admissible during next week’s extradition hearing.
Both the White House and Mr Rohrabaher have previously confirmed some of the key events in the claims made by Assange's lawyers.
One month after his visit to Assange in August, the White House confirmed Mr Rohrabaher had discussed a possible deal for the Wikileaks founder in a call with Mr Trump's chief of staff John Kelly.
Mr Rohrabacher told the Wall Street Journal that as part of the proposed deal, Assange would need to hand over solid digital evidence proving Russia was not behind Hillary Clinton's hacked emails.
It is unclear if Mr Trump and Mr Rohrabacher ever directly discussed Assange. But in Mr Rohrabcher complained to The Intercept that he had been blocked from briefing Mr Trump about his August meeting with Assange because White House officials were scared discussions it could look like collusion.
“Not only Kelly, but others are worried if I say one word to Trump about Russia, that it would appear to out-of-control prosecutors that that is where the collusion is,” Mr Rohrabacher said.
Assange is wanted in America to face 18 charges, including conspiring to commit computer intrusion, over the publication of US cables a decade ago. He could face up to 175 years in jail if found guilty.
He is accused of working with former US army intelligence analyst Chelsea Manning to leak hundreds of thousands of classified documents.
The extradition hearing is due to begin at Woolwich Crown Court on Monday. The decision, which is expected months later, is likely to be appealed against by the losing side, whatever the outcome.
Assange has been held on remand in Belmarsh prison since last September after serving a 50-week jail sentence for breaching his bail conditions while he was in the Ecuadorian embassy.
He entered the building in 2012 to avoid extradition to Sweden over sex offence allegations, which he has always denied and were subsequently dropped.
Assange appeared in court on Wednesday by videolink from HMP Belmarsh. He spoke to confirm his name and date of birth, and sat holding a stack of papers during the hearing.
The hearing came one day after Australian MPs George Christensen and Andrew Wilkie called for Boris Johnson to intervene and stop the extradition hearing.
At a press conference on Tuesday it emerged that fellow inmates at the high-security prison successfully lobbied for his release from solitary confinement.
Speaking after the press conference, Assange’s father John Shipton said his son’s condition had improved, but said the extradition going ahead would be akin to a “death sentence”.
Additional reporting by PA