Kushner, 36, was Trump's main intermediary with foreign governments during the 2016 election campaign and now plays that role in the White House.
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He arranged meetings between Trump and leaders from Japanese Prime Minister Shinzo Abe to Mexican President Enrique Pena Nieto.
But it is his contacts with Russian officials that are now coming under the microscope, amid explosive allegations of collusion between the Trump campaign and the Kremlin.
US intelligence has concluded that Russia launched a broad-ranging campaign designed to help Trump win election.
"Throughout the campaign and transition, Jared Kushner served as the official primary point of contact with foreign governments and officials," a White House official said.
"Given this role, he has volunteered to speak with Chairman Burr's committee, but has not yet received confirmation," the official said, referring to Senator Richard Burr of North Carolina.
Burr chairs the Senate intelligence committee.
In a joint statement with his Democratic counterpart Senator Mark Warner, Burr said Kushner's decision to appear showed the panel's independence.
"From the beginning of this investigation we have committed to follow the facts wherever they lead us," they said.
"Mr. Kushner will certainly not be the last person the committee calls to give testimony, but we expect him to be able to provide answers to key questions that have arisen in our inquiry."
The development comes amid renewed questions over the impartiality of a parallel inquiry from the House of Representative's intelligence committee.
It is led by Congressman Devin Nunes, who is under fire for briefing the Trump about issues related to the investigation.
Last week Nunes revealed that Trump's own communications may have been swept up in intelligence gathering on suspected foreign agents.
Nunes worked on Trump's transition team and is now leading an investigation into possible links between that campaign team and Russia.
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The Senate's top democrat Chuck Schumer called Monday for Nunes to be removed from his chairmanship of the inquiry.
"Chairman Nunes is falling down on the job and seems to be more interested in protecting the president than in seeking the truth," Schumer said on the floor of the Senate.
"You cannot have the person in charge of an impartial investigation be partial to one side."
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