"Number one, we are chasing our tails as a nation when it comes to the Russians", the Republican senator said.
President Trump and his top advisers are reportedly considering creating a White House "war room" and a staff shake-up to combat current and future revelations related to the Russian Federation controversy, The Washington Post reported on Saturday.
Homeland Security Secretary John Kelly said he didn't know if the news reports were true but described back-channel communications as a "good thing".
Economic adviser Gary Cohn said: "We're not going to comment on Jared".
Mr Trump's private legal team, led by his NY lawyer, Mr Marc Kasowitz, was preparing to meet in Washington to face new questions about contact between Mr Kushner and representatives of Russian President Vladimir Putin.
"The White House disclosed the fact of the meeting only in March, playing down its significance".
The story about the alleged phone calls was reported by Reuters, the day after reports about Kushner, Trump's son-in-law, now being part of a federal investigation into Trump associates allegedly colluding with Russian Federation to win the 2016 White House race.
The intent was to connect Trump's chief national security adviser at the time, Michael Flynn, with Russian military leaders, said this person, who wasn't authorized to publicly discuss private policy deliberations and spoke on condition of anonymity. Flynn remains under federal investigation in Virginia over his foreign business ties.
"The authority now for the two leaders means they can require and compel legally more information from other people, and you're starting to see that even though there was a sense of, "Oh, we have time; we're trying to do this right" just a few weeks ago, as everything else kicks into high gear, you're going to see some kicking into high gear on Capitol Hill, too".
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The 36-year-old Kushner, a real estate developer with no previous government experience, had at least three previously undisclosed contacts with the Russian ambassador to the United States during and after the 2016 presidential campaign, seven current and former US officials told Reuters.
"I've heard from senior White House sources that Trump himself has told Jared Kushner, 'Look, people like you, young, wealthy, NY liberal Democrats, are not the reason I'm in the White House, ' Warren reports".
The Times reported that Flynn was at the centre of the idea.
In addition, Reuters reported that Kushner had at least three previously undisclosed contacts with Kislyak previous year, including two phone calls between April and November.
Recently appointed special counsel Robert Mueller, a former FBI director, is starting off an investigation with a broad mandate that will allow him to probe both the possible Russian influence and whether Trump attempted to obstruct the investigation by firing FBI Director James Comey. Kushner is eager to cooperate with any and all inquiries, this official said.
While the FBI is investigating Kushner's contacts with Russian Federation, he is not now a target of that investigation, the current law enforcement official said.
Kushner has already volunteered to speak with Congress about those meetings, and his attorney says he's willing to cooperate with any additional investigations.
The Post and other media have been careful to note that their sources did not say Kushner was a "target" of the investigation, nor that he was accused of any wrongdoing. "Kushner's security clearance should be suspended until the FBI's findings are complete".