The request to a federal court in San Francisco came from the defense team of Pavlo Lazarenko, prime minister of Ukraine from 1996-97, who now faces money-laundering charges.
The lawyers also later filed a second motion to dismiss all charges because they alleged a highly placed U.S. official had leaked information about the Lazarenko trial to Kuchma.
"The relationship between the United States and Ukraine has been such that the abuses of the Ukraine system, the violation of human rights, the manufacturing of information and the sharing of confidential United States information has irreparable [sic] tainted the United States prosecution," attorney Daniel Horowitz wrote.
Convicted in absentia for money laundering in Switzerland and charged with murder in Ukraine, Lazarenko maintains he is the victim of a political vendetta by Kuchma, Ukraine's president since 1994.
The tapes the defense wants the jury to hear come from Kuchma's former bodyguard, Mykola Melnychenko, who secretly recorded hundreds of hours of Kuchma's conversations. Lazarenko's U.S. trial started in March.
"They can prove that President Kuchma has affected this case," Horowitz said.
"How can you have a case when the evidence is tainted and you know it?"
If the judge allows the motion filed on Tuesday morning, Melnychenko would testify after excerpts of the tapes are played.
"I'll comment after the hearing because this is so serious," Melnychenko, who was in San Francisco on Tuesday, said when asked to comment on his possible testimony.
The former bodyguard has been granted exile in the United States.
The defense filing alleges Kuchma went after Lazarenko's allies with false charges to maintain political control over the country.
"The defense is prepared to play a recording which portrays President Kuchma and his manipulation of our judicial system in a light which is repulsive to any person concerned with real justice," the motion read.
"From these recordings it is clear that Lazarenko is being persecuted not because he violated Ukraine law but because he did not follow the lead of Leonid Kuchma."
According to short excerpts of the recordings, the Ukrainian president laced his speech with frequent expletives, and on several occasions uses the word "kike," a derogatory term for Jews.
Parts of the Melnychenko tapes have come to light previously.
In 2002, the United States suspended some aid to Ukraine after saying the tapes showed that Kuchma had breached United Nations sanctions and sold Iraq an early warning system.
In 2001, the tapes appeared to link the president to the murder of Ukrainian journalist Georgy Gongadze.
Attorney Horowitz said the tapes pertinent to the Lazarenko trial have not been made public to date.
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