WASHINGTON—Lev Parnas, an associate of Rudy Giuliani, said he warned a top aide to the newly elected president of Ukraine that the U.S. would halt aid to the country if it didn’t announce investigations that could benefit President Trump politically.

In a May meeting with Serhiy Shefir, an aide to then-Ukrainian President-elect Volodymyr Zelensky, Mr. Parnas delivered the message at the direction of Mr. Giuliani, Mr. Trump’s personal lawyer, he said in an interview with MSNBC’s Rachel Maddow on Wednesday.

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