According to USA officials who intercepted and reviewed Russian communications regarding the proposal, Kislyak told his superiors that Kushner, Trump's son-in-law and senior adviser, suggested the communications line during a meeting at Trump Tower on December 1 or 2.
President Trump's son-in-law Jared Kushner and the Russian ambassador talked about setting up a secret communication link between Trump's transition team and the Kremlin, according to published reports.
The Post previously reported that investigators are focusing on meetings he held in December with Kislyak and the head of a Russian bank that has been under USA sanctions since 2014.
McMaster told American reporters traveling with Trump at the G7 summit in Sicily that the USA frequently sets up backchannels to countries for discreet communications.
The White House declined to comment as did the Russian Embassy. A White House official confirmed plans to hold more rallies.
Russian Federation at times feeds false information into communication streams it suspects are monitored as a way of sowing misinformation and confusion among US analysts.
The Post's report capped off a bad week for Kushner.
The Russian Foreign Ministry dismissed reports that Kushner asked Kislyak about establishing a secret means of communication with Moscow.
To some officials, it also reflects a staggering naivete.
In February, Mr. Flynn had to resign as national security adviser for misleading Vice President Pence about the nature of phone calls he had in late December with the Russian ambassador.
"How would he trust that the Russians wouldn't leak it on their side?" said one former senior intelligence official.
Initially on Saturday, McMaster referred the first question about Kushner to White House press secretary Sean Spicer. The calls focused on how to improve economic relations between the United States and Russian Federation, and how the two countries could better cooperate in the fight against Islamist extremism, Reuters reported, citing six current and former American officials.
Flynn stepped down after lying about conversations he had with Kislyak.
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. The bank has been under US sanctions since 2014.
Despite Trump's repeated denials that his team colluded with Russian Federation, former CIA director John Brennan told the Senate Intelligence Committee on May 23 that the investigation into possible collusion between Trump campaign officials and the Kremlin is "well-founded".
"We have backchannel communications with any number of individual [countries]. what that allows you to do is communicate in a discreet manner".
This person says the administration felt there was no need for a back channel once Secretary of State Rex Tillerson was confirmed, and the Trump team chose to communicate with Moscow through more official channels.
Trump's advisers were similarly secretive about meetings with leaders from the United Arab Emirates.
Reuters corroborated the allegation in its own report late Friday and added that Mr. Kushner had at least three previously undisclosed contacts with Mr. Kislyak during and after Mr. Trump's 2016 presidential campaign, citing seven current and former US officials.
He said the Trump administration at the senior levels was being "consumed by its own hubris" in how it engaged with global actors. Doing so would require Moscow to expose its most sophisticated communications capabilities - which are likely housed in highly secure locations at diplomatic compounds - to an American.
The Post's anonymous sources said that Kislyak was even taken aback by the suggestion.
On May 9, President Trump fired FBI Director James Comey, who helmed a probe into the Trump campaign's ties with Russian officials.
At the time, Mr Kushner was a private citizen, but working in Mr Trump's transition team.