White House officials defended the concept of secret communications channels without commenting specifically on the Kushner case.
Trump, meanwhile, resumed his Twitter postings from the White House on Sunday and dismissed any story based on anonymous sources.
"Leaks are an issue", Schiff said.
Former Director of National Intelligence James Clapper, in an interview on NBC's "Meet the Press", said he would not confirm the stories about Kushner's conversations, but added that he was "very concerned" about reports of the Trump administration approaching Russian Federation "from a theoretical standpoint".
Today, Kelly defended Kushner, who is being investigated by the Federal Bureau of Investigation over his contacts with Russian officials, saying that 'any communication to a country, particularly a country like Russia, is a good thing'.
"We're chasing our tails as a nation when it comes to the Russians", Graham said in CNN's "State of the Union" on Sunday. The newspaper cited anonymous USA officials who were briefed on intelligence reports on intercepted Russian communications.
"The FBI's Russian Federation investigation reached Trump's backyard, and now it's in his house", the DNC Deputy Communications Director Adrienne Watson said in a statement Thursday.
According to the person familiar with the Kushner meeting, the Trump team eventually felt there was no need for a back channel once Rex Tillerson was confirmed as secretary of state on February 1.More news: Irfan Pathan backs India to defend Champions Trophy title
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Flynn served briefly as Trump's national security adviser before being fired in February after officials said he misled Vice-President Mike Pence about whether he and the ambassador had discussed us sanctions against Russian Federation in a phone call.
The Post said the Russian ambassador to the United States, Sergei Kislyak, was surprised by the future White House aide's idea of a secret channel and passed it on to the Kremlin.
The White House crisis management plan, Jared Kushner's standing with the boss and new congressional wrinkles in the Russian Federation investigation - they're all a part of our Inside Politics forecast.
But CNN's Jeff Zeleny, fresh back from covering the trip, said Kushner was aware of a coming Washington Post report detailing his role in the Russian Federation election meddling investigation and decided it was best not to be at the President's side on the world stage when that news broke.
Kelly on Sunday also said he didn't take issue with a report, originally published in the Washington Post, that indicated White House Senior Adviser Jared Kushner, also President Trump's son-in-law, proposed a secret communication channel with the Kremlin so that discussions before the inauguration could not be detected.
"But for people associated with the campaign after that campaign has ended and where the Russians during that campaign were helping you, to try to establish a back channel and hide it from your own government, that's a serious allegation", he said.
The administration had hoped Trump s nine-day turn in the worldwide spotlight would offer a respite from relentless coverage of an ever-widening probe into Russian meddling in a year ago s election and possible collusion by the Trump campaign - with its endless barrage of newspaper scoops and explosive revelations.
The Illinois Democrat also called the reports "disturbing" and demanded that Kushner testify before the Oversight Committee.
Kushner's lawyer said he "has no recollection of the calls described", and that they have requested dates of the alleged undisclosed contacts to they can investigate.
The request from the committee arrived last week at campaign headquarters in NY, according to person familiar with the request who wasn't authorized to discuss the developments publicly and spoke on condition of anonymity.