Ukrainian President Volodymyr Zelenskiy has after he was caught on tape saying Mr Zelenskiy - a former sitcom star - knows nothing about the economy.
In a video released by Mr Zelenskiy's office, the president called the situation "unpleasant" but said he had decided to "give a chance" to Prime Minister Oleksiy Honcharuk and his cabinet.
The furore comes at a fraught moment for Mr Zelenskiy, who has found himself in the middle of the impeachment case
Mr Trump stands accused of withholding nearly $US400 million ($A582 million) in military aid to Ukraine to pressure the country's leader to investigate his political rival, Joe Biden.
In a Facebook post earlier on Friday, Mr Honcharuk praised Mr Zelenskiy as "an example of transparency and decency to me".
"In order to dispel any doubts about our respect and trust for the president, I have written a resignation letter and submitted it to the president for introduction to parliament," he wrote.
Ukrainian Prime Minister Oleksiy Honcharuk offered his resignation. Source: EPA
The offer to step down was subject to approval by the Rada, Ukraine's parliament. But analysts expressed doubt the resignation would come to pass.
"Zelenskiy doesn't want to dismiss Honcharuk," Penta thinktank head Volodymyr Fesenko said.
Razumkov Centre political analyst Volodymyr Sidenko said Ukraine's president fears prompting a political crisis in the country and doesn't want to complicate his relationship with foreign investors and the International Monetary Fund.
"Honcharuk's resignation can destroy the idea of the government's unity and cast a doubt on Zelenskiy's ability to control the situation," he said.
Earlier this week an audio recording surfaced in which Mr Honcharuk appeared to make disparaging comments about Mr Zelenskiy's understanding of economics.
Donald Trump meets Ukrainian President Volodymyr Zelensky on the sidelines of the UN General Assembly. Source: AP
He called the president a layman" in economics and said the president should be better educated about the national currency.
Mr Zelenskiy is a 41-year-old former comedian whose only political experience before his election last spring consisted of playing a Ukrainian president on TV.
