The Washington Post reported last week that Kushner and Russian Ambassador Surgey Kislyak met at Trump Tower in December to discuss the possibility of setting up secret, back-channel communications between the President's transition team and officials in Moscow.
Trump did not come out directly and defend Kushner, but decried what he called the "fake news media" in a series of tweets earlier Sunday.
The White House faces mounting questions about potential ties between Russian Federation and Mr Trump's presidential campaign, which are also the subject of criminal and congressional investigations.
Stephen Miller, senior adviser to President Donald Trump, left, and National Security Adviser H.R. McMaster walk from Marine One across the South Lawn to White House in Washington, Saturday, May 27, 2017, as they return from.
"If these allegations are true and he had discussions with the Russians about establishing a back-channel and didn't reveal that, that's a real problem in terms of whether he should maintain that kind of a security clearance", Schiff said. But Kelly said such back-channel communications don't bother him and would not be harmful to USA security interests.
The White House is not always troubled by anonymous sources.
The administration had hoped Trump's nine-day turn in the worldwide spotlight would offer a respite from relentless coverage - marked by an endless barrage of explosive revelations - of an ever-widening probe into Russian meddling in last year's election and possible collusion by the Trump campaign.
"When you see stories get perpetrated that are absolutely false; that are not based in fact, that is troubling, and he is rightly concerned", Spicer said.
Kushner is being scrutinized in the agency's investigation into Russian meddling in the USA election over his meetings with at least two Russian officials, Moscow's ambassador to the US and a banking executive, sources have told ABC News.
Mr Kushner initially had come to the attention of FBI investigators past year as they began scrutinising former national security adviser Michael Flynn's connections with Russian officials, two sources told Reuters. "What that allows you to do is communicate in a discrete manner". He said there was also a big question as to what role Mr Kushner should have. "We're chasing our tails as a nation when it comes to the Russians", Senator Lindsey Graham told CNN.
Even as he did not directly answer questions from reporters, Spicer nearly seemed to confirm stories that Kushner had sought a secret communications channel with Moscow during the transition.
Asked if such communications were a big deal, Kelly said: "any time you have channels of communication with a country, particularly one like Russian Federation, I wouldn't criticize it".
"My dashboard warning light was clearly on", Clapper said. "They are in to do us in".