HR McMaster, the national security adviser, was also dispatched to defend Mr Kushner, saying: "we have back-channel communication with a number of countries".
Kushner's lawyer has also publicly said the president's son-in-law is "willing to cooperate in the probe". He is not suspected of a crime, however, or expected to face charges.
CNN, meanwhile, reported on Tuesday that "Russian government officials discussed having potentially "derogatory" information about then-presidential candidate Donald Trump and some of his top aides in conversations intercepted by United States intelligence during the 2016 election". "He is respected by virtually everyone and is working on programs that will save our country billions of dollars".
The new report follows allegations last week that Mr Kushner and the Russian ambassador to America, Sergey Kislyak, discussed setting up a secure, private line of communication with Moscow using Russian facilities that bypassed the USA government.
The CNN report, quoting "two former intelligence officials and a congressional source", said the derogatory information was "financial in nature".More news: French Open 2017: Nadal, Djoko ease past Round One
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"My dashboard warning light was clearly on and I think that was the case with all of us in the intelligence community - very concerned about the nature of these approaches to the Russians", Clapper told NBC's "Meet The Press".
After maintaining a limited social media presence throughout his trip, Trump on Sunday unleashed a furious flurry of tweets, lashing out at what he called the "fake news" media.
That raises big questions about the White House's claims that the meeting was part of an effort to establish a way of communicating with the Russians outside the normal channels that would possibly be monitored by USA intelligence - an effort that in itself is highly unusual.
Kushner's connection with the Russian Federation probe was alleged in a Washington Post report while Trump was on a nine-day trip to the Middle East and Europe.
CNN political analyst Chris Cillizza also said he doesn't believe Kushner was "lone-wolfing" it at the time, but was following orders from his father-in-law or Michael Flynn, who would later be booted as Trump's national security advisor over his ties to Russian Federation. One key lawmaker, Senator Bob Corker, chairman of the Senate Foreign Relations Committee, said Sunday, "He seems to be a very open person".