The White House on Monday insisted Kushner's meetings were appropriate given his broad profile during the presidential campaign and transition.
"Throughout the campaign and transition, Jared Kushner served as the official primary point of contact with foreign governments and officials. Given this role, he has volunteered to speak with (Senate Intelligence) Chairman (Richard) Burr's Committee, but has not yet received confirmation," a White House spokesperson told CNN.
The Senate Intelligence Committee put out a statement from Burr and Vice Chairman Mark Warner confirming they were aware of Kushner's offer and said details were still being hammered out.
"The timing of Mr. Kushner's testimony is still being determined, but will only come after the committee determines that it has received any documents or information necessary to ensure that the meeting is productive for all sides," the joint statement said.
The New York Times first reported
Monday that Kushner was wanted for questioning because of his role in arranging meetings between campaign advisers and Kislyak, along with other foreign leaders.
In December, during the presidential transition, Kushner met with the Russian ambassador at Trump Tower.
Following the meeting, Kislyak requested another sit down. Instead of attending himself, Kushner sent his deputy, Avrahm Berkowitz, a senior administration official confirmed.
Kushner had one additional meeting with a Russian official, according to a senior Trump administration official. This one took place between Kushner and Sergey N. Gorkov, the head of Russia's economic development bank, at the insistence of the Russian ambassador.
On Monday, the White House addressed the reports, but its description of Kushner's meetings came at odds with the description Gorkov's bank offered.
White House press secretary Sean Spicer described Kushner as a "conduit" during the transition period for leaders around the world to establish their bearings with the incoming administration.
"That's until we had a State Department, a functioning place for people to go," Spicer said.
Vnesheconombank, in a statement, described the meeting as a business meeting, saying bank management met "with a number of representatives from the largest banks and business circles in the United States, including the head of Kushner Companies, Jared Kushner."
Last week, White House spokeswoman Hope Hicks downplayed Kushner's meetings with Russian officials, noting that Kushner met only once with Kislyak.
"Jared's only meeting with Ambassador Kislyak took place during the transition, as has been reported. He met hundreds of people on the campaign and may have shaken the Ambassador's hand prior to an April 2016 speech on foreign policy by the President," Hicks told CNN last Wednesday.
Kushner was involved in a wide array of issues during the presidential transition and maintains a broad scope of influence in the White House. In that context, his meetings with Russian officials may have been entirely appropriate.
However, the meetings are drawing increased scrutiny as committees in the House and Senate -- as well as FBI investigators -- are looking at the extent of Russia's meddling in the 2016 election and potential ties between Trump associates and suspected Russian operatives.
The White House has struggled to move beyond questions about Russia, in part because previously undisclosed contacts between Trump associates and Russian officials have repeatedly come to light.
Michael Flynn, Trump's former national security adviser, was also in attendance for that December meeting at Trump Tower with the Russian ambassador and Kushner. Flynn was ousted in February after misleading senior administration officials about his contacts with Russian officials.