(CNN) Senate Intelligence Chairman Richard Burr said his panel won't share with another key committee a transcript of an interview his investigators conducted with Jared Kushner, a move that could force President Donald Trump's son-in-law to return to Capitol Hill for further questioning.

Burr told CNN that the transcript of the July interview won't leave his committee, despite a request from a second panel, the Senate Judiciary Committee, to review Kushner's responses behind closed doors to Senate Intelligence as part of its Russia investigation.

"We don't hand out transcripts," Burr said in the Capitol. "No."

Moreover, the Senate Intelligence Committee is also signaling it wants to bring Kushner back for further questioning as well.

"I believe there are an awful lot of questions that Mr. Kushner still has to answer," Sen. Mark Warner, the top Democrat on Senate Intelligence, told CNN.

The pressure on Kushner comes after the leaders of the Senate Judiciary Committee suggested that he had not been responsive to the committee's requests, demanding to see transcriptions of his interviews with other panels.

The comments are significant because the two leaders of the Senate Judiciary, Sens. Chuck Grassley and Dianne Feinstein, told Kushner in a letter earlier this month that it wanted to review transcripts from his interviews with other congressional committees, including the House and Senate Intelligence panels, if he did not agree to come before Judiciary for questioning.

The refusal to share the transcript between committees also points to another lingering issue on Capitol Hill: That the committees often butt heads as they compete for witnesses and information as part of the three separate Hill investigations into Russia meddling.

"Those committees have not provided us with transcripts of any of their other witness interviews," Grassley and Feinstein said in their letter to Kushner. "If you are able to secure for the Committee copies of the transcripts from Mr. Kushner's other interviews, then please provide them and we will consider whether the transcript satisfies the needs of our investigation."

In response to the letter, Kushner's attorney noted that his client had already spent "six or more hours answering any relevant question" to Senate and House Intelligence and said they are not in possession of the transcript.

"I do not understand why these committees would not provide the transcripts to you," wrote attorney Abbe Lowell in the November 17 letter to Grassley and Feinstein. "We have told both committees we have no objection to their providing them to you or in response to any bipartisan request from a committee with proper jurisdiction."

In an interview with CNN last week, Lowell said his client is willing to cooperate if another interview is sought.

"Mr. Kushner has been very clear that will cooperate as he has been voluntarily with all bipartisan requests from committees on anything that's relevant. He's done it and he'll do it again," Lowell told CNN's Evan Perez on November 19.

Lowell did not respond to CNN request for further comment.

The Senate Judiciary Committee initially set a Monday deadline for Kushner to provide a wide-range of information, asking for records regarding communications with Trump's former national security adviser, Michael Flynn, as well as a correspondence about WikiLeaks during the campaign season and information about a security clearance form that initially failed to disclose his foreign contacts. But that timeline demand has eased as both the Judiciary panel and Kushner's team are engaged in negotiations for Kushner to provide the documents the committee requested.

Grassley told CNN Monday that Kushner was being "cooperative" to the panel's latest requests as part of the latest round of talks.

Asked if he still wanted to bring Kushner in for a transcribed interview, Grassley said he wanted to see what documents the White House adviser first provides to the committee.

"I think what we're after is the documents that evidently we didn't make clear in our first request that we wanted," Grassley said. "We are exchanging letters on that point. We're going to review that stuff before I can answer the question."

Despite meeting with two Hill panels in July, Kushner remains of interest on Capitol Hill because of his contacts with Russians during the campaign season and transition as well as the June 2016 meeting at Trump Tower, which was also attended by Russian operatives and the President's eldest son, Donald Trump Jr.

All three Hill panels still want to question Trump Jr. this year. Trump Jr. has already met once this fall with Senate Judiciary staff.

"We plan to interview him," Burr told CNN, referring to the President's son. "I don't know when that will happen. ... We don't have that set yet."

This story has been updated and will continue to be updated with new developments.