The Washington Post said Kushner's secret communications proposal was made Dec 1 or 2 at Trump Tower in NY, according to intercepts of Russian communications that were reviewed by USA officials.
A White House official close to Kushner pushed back on any notion that Kushner would take leave from his West Wing duties in light of recent reports.
The Post says Kislyak was reportedly "taken aback" by the suggestion. The White House did not respond to a request for comment.
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.
At the time, Mr Kushner was a private citizen, but working in Mr Trump's transition team.
Current and former USA intelligence officials said that although Russian diplomats have secure means of communicating with Moscow, Kushner's apparent request for access to such channels was extraordinary. "The aides say Jared Kushner is going to remain, keep his head down, but this raises more questions about him".
The entire idea, he said, "seems extremely naive or absolutely insane".
Trump returned late Saturday from his first global trip as U.S. president, geared up to combat concerns over aides' ties to Russian Federation including explosive reports that his son-in-law Jared Kushner sought a secret communications line with Moscow. He added: "It allows you to communicate in a discreet manner".
Underdog Cavs insist they have plenty of bite for Finals
We didn't talk about championships or anything. "With this team , the center position is one that we're looking for", he said. About an hour after the final buzzer sounded, James took a seat at the interview podium, a black cap pulled low on his brow.
National Security Adviser HR McMaster said backchannel communications between different governments was normal.
The Washington Post had a story last night on Jared wanting to establish a back channel with the Russians during the transition that adds detail-he talked to Kislyak about it and allegedly wanted to use Russia diplomatic facilities for the conversations-to what was already known.
The FBI, in the course of its investigation into Russian involvement in the 2016 election, is now also looking at Kushner's multiple roles in Trump's campaign operation and his contacts with Russian officials. "The danger is that they are going to care just about re-election, and not passing any bill". One White House official at the time characterized it as a brief courtesy meeting and confirmed that Michael Flynn, Trump's ousted national security adviser, was in the room. Trump stepped up to insist he'd never said the word "Israel" in his talks with Russian officials-effectively confirming that nation as the source of the information.
Minutes after Trump and his entourage left Washington for Saudi Arabia on 19 May, two news stories broke: that the president had described recently fired Federal Bureau of Investigation director James Comey as a "nut job" during his meeting with Russia's foreign minister and ambassador to the U.S., and that an unnamed White House official was the target of Federal Bureau of Investigation scrutiny.
Another potential line of inquiry could concern Kushner's failure to disclose some of his contacts with Russian government officials when he was filling out his application for a security clearance.
The news of the secret channel proposal came in the same week that Reuters reported that Kushner had three unreported contacts with the Russians between April and November of a year ago.
Regarding Kushner, former FBI agent Jim Treacy said Friday: "If there is an investigation on anybody, would other folks around that person be of interest to the FBI as far as being interviewed?" When proposed by president-elect's son-in-law using Russian comms w/ no clear national security goal.