Kushner, a close adviser to Trump, went so far as to suggest using Russian diplomatic facilities in the United States to protect such a channel from being monitored, The Post said, quoting US officials briefed on intelligence reports.
The revelation is yet another sensational element in the deluge of allegations raising questions about the Trump team's relationship with the Russians, who US intelligence agencies say tried to sway the November election in Trump's favor and thus deny Hillary Clinton the presidency.
And it ensures that Trump will be thrust back into the din of the Russia scandal upon his return to Washington this weekend following his first foreign trip, a tour of the Middle East and Europe.
The Washington Post said the secret communications proposal was made December 1 or 2 at Trump Tower in New York, according to intercepts of Russian communications that were reviewed by US officials.
Michael Flynn, who would become Trump's national security adviser before being fired 24 days into the job for not telling the truth about meetings he held with the Russian ambassador, was also at the meeting, The Post reported.
The Russian ambassador, Sergei Kislyak, was surprised by Kushner's idea of the secret channel and passed it on to the Kremlin, the Post said. It did not specify what came of Kushner's alleged pitch, if anything.
The White House did not immediately comment on the report.
Jim Himes, a Connecticut Democrat on the House Intelligence Committee, told MSNBC, that not disclosing such contacts "raises a lot of questions."
"If it's true that somebody did ask for some kind of private line, some kind of mode of communication that was secure, that was unusual, boy does that raise a whole bunch of questions."
Besides the Kushner developments, which strikes at Trump's core by drawing his family into the crisis, the White House also faces a cascade of other worries in the coming week.
The Senate Intelligence Committee has "asked President Trump's political organization to gather and produce all Russia-related documents, emails and phone records going back to his campaign's launch in June 2015," The Post reported. There was no immediate White House reaction.
Fired former FBI director James Comey has promised to testify at an open session before the Senate Intelligence Committee, sometime after Monday's Memorial Day holiday.
The White House staff itself could be facing upheaval. CBS News reported that Trump is expected to consider plans for a shakeup of his communications operation upon his return.
And Oleg Deripaska, a Russian once close to Trump's former campaign manager, has offered to cooperate with congressional bodies probing alleged Russian meddling in the 2016 US election, The New York Times reported Friday.
Kushner at center stage
But Kushner, a 36-year-old real estate developer who is married to Trump's daughter Ivanka, is likely to take center stage in the coming days.
Reserved in public, he was on prominent view during Trump's first presidential trip, as befits a trusted behind-the-scenes adviser involved in everything from Middle East peace to an initiative to streamline the US bureaucracy.
The Post reported earlier that investigators are focusing on meetings he held in December with Moscow's ambassador and the head of a Russian bank that has been under US sanctions since 2014.
Kushner has offered to talk to Congress about these meetings, according to his lawyer Jamie Gorelick.
The Post and other media have been careful to note that their sources did not say Kushner was a "target" of the investigation, nor that he was accused of any wrongdoing.
If he were a "target," it would suggest Kushner was a main suspect of the investigation.
The Post reported last week that the Russia investigation had been extended to a top White House official as a "significant person of interest."
Kushner is the only person currently in the White House known to be under investigation.
At least four other former campaign aides or advisers have been reported to be under FBI scrutiny as well -- Flynn, former campaign manager Paul Manafort, sometime Trump adviser Roger Stone, and ex-campaign adviser Carter Page.
The FBI investigation is now being overseen by Robert Mueller, a respected former FBI director who was given broad powers to pursue the case as a special counsel after Trump abruptly fired Comey on May 9.
The key question before the FBI is whether the Trump campaign colluded with Russia in its effort to tilt the 2016 US election in the Republican's favor, which included a damaging hack of Democratic campaign emails.
Trump has denied any collusion, calling the probe "the greatest witch hunt" in American political history.
Former CIA director John Brennan revealed this week that intelligence chiefs had been looking into suspicious contacts between Trump campaign associates and Russian officials since mid-2016.
The Senate and House Intelligence Committees also are investigating, but not with an eye to bringing criminal charges.
In early December, after Trump had won the elections, Kushner and Flynn met in New York with Kislyak.
Kushner also met that month with Sergei Gorkov, chairman of VneshEconomBank, the state bank under US sanctions since July 2014.
Both those meetings have since been publicly acknowledged by the White House, but Kushner initially failed to declare them on forms submitted to obtain a security clearance.
His lawyer later said it was a mistake, telling the FBI that he would amend the forms. Not making a full and honest statement can be a federal crime punishable of up to five years in prison.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)