Ukraine President Volodymyr Zelensky (L) listens as U.S. President Donald Trump speaks during the 74th Session of the United Nations General Assembly at UN Headquarters in New York, September 24, 2019.

Ukraine's president is coming under increasing pressure to reveal what was said during a much-reported phone call with President Trump but is so far keeping tight-lipped about the exchange just as the Democrats launch an impeachment inquiry against the Trump.

On Wednesday, President Zelensky was asked about his now infamous July phone call with Trump, in which it's alleged that the U.S. president put pressure on Ukraine to investigate former Vice President Joe Biden and his son, Hunter, ahead of the 2020 election.

Trump denies the allegations and has authorized a transcript of the call to be released Wednesday. Meanwhile, Ukraine President Zelensky responded by said that nobody can put pressure on him except his six-year-old son.

"Nobody can put pressure on me because I am the president of an independent state," Zelenskiy told reporters in New York, in comments broadcast by Russia 24, and reported by Reuters, on Wednesday ahead of Zelensky's meeting with Trump.

"The only one person by the way who can put pressure on me ... is my son, who is six years old."

Trump and Zelensky are also due to meet on the sidelines of the UN General Assembly in New York Wednesday afternoon. There is intense media attention on Zelensky – a politically-inexperienced president who took office in May after campaigning on an anti-corruption manifesto - back home in Ukraine too.

The online version of the Kiev Post newspaper is covering the story extensively and has headlined its news website with comments the president made to Voice of America's Ukrainian Service Wednesday, saying "We want the U.S. to support Ukraine."

Meanwhile, news site Dzerkalo Tyzhnia carried an article on Wednesday in which it previews the Trump-Zelensky meeting and says the Ukraine president plans to invite Trump to Ukraine when they meet. Another Ukrainian newspaper KP led with a story on what the world's press is saying about Zelensky in the context of the allegations.

Zelensky might have made light of pressure on him but the phone call between him and Trump has proved politically explosive.

It's alleged that Trump pressured Ukraine for information that could smear Democratic presidential candidate Biden, and damage his chances in the 2020 election race, in exchange for the release of $400 million in aid.

House Speaker Nancy Pelosi on Tuesday said the House of Representatives will launch a formal inquiry into whether President Donald Trump should be impeached.

There's a long way to go before Trump faces real danger, however. Only three American presidents before Trump have faced serious impeachment proceedings, and Congress has never removed one from the White House. Even if Democrats eventually impeach Trump, the Republican-held Senate may never find him guilty and remove him from office.

Close watchers of the new Ukrainian president say he's in a tough situation, particularly as Ukraine is a vulnerable country economically and politically. It is currently trying to negotiate further aid from the International Monetary Fund (IMF) and is dealing with a powerful Russia to the east, with an ongoing conflict with pro-Russian rebels in the Donbass region.

Timothy Ash, a senior emerging markets strategist at Bluebay Asset Management, said in an email Tuesday that "Zelensky finds himself in an impossible position caught between the GOP (Republican Party) and the Democrats, and falling foul or whoever wins the POTUS election in 2020. Best to remain shtum, as I think the Ukrainian side have been trying to," he added.

Attention is now focused on the release of the transcript of the phone call. Asked about the phone call by CNN on Tuesday, Zelensky said his conversations with Trump were "private and confidential." CNBC has contacted President Zelensky's office for further comment on the phone conversation between the presidents but no one was immediately available.

- CNBC's Jacob Pramuk contributed reporting to this story.