“Biden went around bragging that he stopped the prosecution so if you can look into it… It sounds horrible to me.”
Mr Zelensky says in response: “We will take care of that and we will work on the investigation of the case.”
Thanking Mr Trump, Mr Zelensky says he stayed in Trump Tower in New York City during a previous visit to the US.
On the call, the US president also asks Mr Zelensky to work with US Attorney General William Barr and Mr Trump’s personal lawyer, Rudolph Giuliani, to look into the matter, according to the notes.
The Department of Justice said on Wednesday that Mr Trump had not spoken with the attorney general about having Ukraine investigate Mr Biden, and Mr Barr had not communicated with Ukraine.
What is the claim against Joe Biden?
Mr Trump and his conservative allies have focused on how Mr Biden, as US vice-president in 2016, lobbied Ukraine to fire Mr Shokin.
Mr Shokin’s office had opened an investigation into Burisma, a natural gas company on which Mr Biden’s son, Hunter Biden, was a board member.
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Other Western officials had called for Mr Shokin to be fired because of the perception that he was soft on corruption.
Mr Biden last year told a foreign policy event how he threatened to withhold $1bn in aid to Ukraine unless Mr Shokin was removed.
No evidence has come to light so far of wrongdoing by the Bidens.
How the controversy unfolded
– 18 July – President Trump orders White House aide to hold back almost $400m in military aid to Ukraine, report US media
– 25 July – President Trump speaks with Ukraine’s leader in a 30-minute phone call
– 9 September – Congress learns of a whistleblower’s complaint about the call, but is blocked by the Trump administration from viewing it
– 11 September – Military aid for Ukraine is cleared for release by the Pentagon and US Department of State
– 23 September – Trump confirms he withheld Ukrainian aid, saying it was due to concerns about “corruption”
– 24 September – Trump says the aid was withheld so that other countries would pay more