"So, now it's time to start officially releasing the transcripts of all witness interviews we have done related to that meeting," Grassley said in the statement.
US media reported last July that the President's son, Donald Trump Jr., had met during the 2016 presidential campaign with Russian attorney Natalia Veselnitskaya expecting to receive confidential information about his father's political rival Hillary Clinton.
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However, Trump Jr. said the conversation with Veselnitskaya was focused solely on the issue of US nationals adopting Russian children.
Grassley said the Judiciary Committee's chances to interview Kushner have been shot, but expressed hope Kushner may be interviewed by the US House of Representatives Intelligence Committee.
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The process of Judiciary Committee questioning witnesses in the Trump Tower meeting has been completed, Grassley added.
Russia has repeatedly denied allegations of collusion with the Trump campaign — and meddling in the US election — and has called such allegations absurd.