It emerged that Mr Trump had withheld USA military aid to Ukraine, but Mr Zelenskiy told journalists in a 10-hour "press marathon" in a Kyiv food court on Thursday that he did not know about the delayed aid until later.
He said he had "several calls" with Trump, but bristled at repeated questions about their relationship.
"I don't want to interfere in any way in the elections" in the USA, he said.
The elder Biden pressured Ukrainian President Petro Poroshenko in 2016 to oust a top prosecutor who was probing Burisma, an energy company that paid Hunter Biden as a board member from 2014 to 2019.
At a rally in Minnesota last night, Mr Trump said the investigation was a brazen attempt to overthrow the Government and would result in a political backlash against Democrats at the ballot box. "This was not the subject of our conversation", Zelenskiy stated about his name with Trump.
"They blocked this money and nobody asked us [for] anything", Zelensky said on Wednesday, referring to the aid, which was eventually delivered.
The White House, with the belief that the Democrats' impeachment efforts are unconstitutional, refused to cooperate and sent an eight-page letter to House Speaker Nancy Pelosi (D-Calif.) on Tuesday addressing the issue. "We are an independent country, we have relations with many countries", not just the U.S.
Russian Federation has said the pullback needs to take place before a summit scheduled for this month between Putin, Zelenskiy, French President Emmanuel Macron and German Chancellor Angela Merkel.
While Mr Zelensky was speaking, prosecutors in the United States said two of Mr Giuliani's associates had been arrested in a scheme to illegally funnel money to a pro-Trump election committee and other USA political candidates. Zelenskiy appears to be playing to both United States political camps to ensure Ukraine has continued support no matter who wins the presidential election next year.
Congressional Democrats accused Trump of holding up military aid to use as leverage to pressure Ukraine for personal and political interests, such as coercing Ukraine to investigate political rival Joe Biden before the 2020 presidential election, but Trump denied all these allegations.
A TV and film comedian, Zelenskiy overwhelmingly won the presidency in April on promises to fight corruption and end the five-year conflict with Moscow-backed separatists in eastern Ukraine.
Volodymyr Zelenskiy's aptly named "media marathon" in a Kyiv food court Thursday started over breakfast croissants.
At the same time, Trump is also pushing Ukraine to investigate any potential wrongdoing by the Bidens.
He told reporters Thursday that the main goal of his conversation with Trump was to set up a meeting with the American president and show reform measures were in place in Ukraine.
"We are now looking again at all cases that were closed or broken up or were investigated earlier to make a decision to reconsider those instances where illegal procedural decisions were made", he said.
Trump responded to Zelensky's comments as well, saying on Twitter that he had done "nothing wrong".