Bolton detailed the August 2019 conversation in an unpublished manuscript for an upcoming book from Bolton which was given to the White House for a standard pre-publication review.
ABC News has not independently reviewed the manuscript.
The reported conversation is the latest claim to emerge about Trump's alleged misconduct surrounding Ukraine since his impeachment in December and comes on the eve of the Trump team starting the heart of its defense in his Senate trial. It also comes as Democrats and others have been pushing for the Senate to call witnesses.
Trump flatly denied the report just after midnight Monday, tweeting, "I NEVER told John Bolton that the aid to Ukraine was tied to investigations into Democrats, including the Bidens. In fact, he never complained about this at the time of his very public termination. If John Bolton said this, it was only to sell a book."
Bolton and other current and former White House aides have been barred by the White House from testifying, although Bolton said he would testify under subpoena. What Bolton would say has been the source of much speculation.
"The draft manuscript was transmitted to the White House for pre publication review by the NSC, the Ambassador has not passed the manuscript to anyone that goes to the reporting in the New York Times," Sarah Tinsley, a spokesperson for Bolton, who left the administration in September 2019, said in a statement on Sunday in response to report. "There is only one hard copy and it was sent to the NSC for review several weeks ago."
Four Republicans would need to break ranks and vote with Democrats in order for witnesses to be called, and the report has only added fuel to the Democrats' call for witnesses.
Republicans have argued that Democrats should have waited for litigation to play out that could have forced Trump's top aides to testify, but Democrats rejected that argument, saying that would have allowed Trump to run out the clock.
During their impeachment arguments on Saturday, the president's legal team stated there is "simply no evidence anywhere that President Trump ever linked security assistance to any investigations."
At issue is nearly $400 million in security assistance to Ukraine that was abruptly put on hold just over an hour after Trump spoke with with the country's new president on July 25, 2019 and asked him to look into the Bidens. The aid was eventually released, but Trump's actions formed the basis of his impeachment.
Trump and his allies have maintained that it was proper for them to inquire about potential corruption on the part of the Bidens - allegations that are still unproven.
The Democratic House managers in the impeachment trial repeated their calls for witnesses on Sunday night.
"Senators should insist that Mr. Bolton be called as a witness, and provide his notes and other relevant documents." the managers said in a statement. "The President knows how devastating his testimony would be."
Rudy Giuliani, the president's personal attorney and a central figure in the impeachment trial, refuted Bolton's claims as reported by the Times in a statement to ABC News on Sunday.
"He never once expressed concern to me," Giuliani said of Bolton in the statement. "If he had confronted me, I could have explained it to him."
"While Bolton was trying to figure out who to invade or unsuccessfully overthrow, Ukrainians and Americans stole almost anything not nailed down," Giuliani continued. "I used to like and respect John and tell people they were wrong about how irresponsible he was. I was wrong."
The upcoming book also includes allegations about Attorney General William Barr, the Times reported. The book states Bolton called Barr after Trump's July 25 phone call with Ukraine President Zelenskiy in order to raise concerns about Giuliani, and inform the attorney general the president had mentioned him on the call.
On Sunday night, a senior DOJ official disputed that Bolton told AG Barr about the Trump-Zelenskiy call or Trump’s repeated mentions of Barr during their conversation, but the official confirmed that Bolton had spoken with Barr following the call to express concerns about Rudy Giuliani, the president's personal attorney, and his activities concerning Ukraine.