Speaking generally, national security adviser H.R. McMaster said "we have back channel communications with a number of countries". When asked if it would concern him, though, if someone in the administration tried to set up a back channel with the Russian embassy or the Kremlin, McMaster replied "no".
The Post's report capped off a bad week for Kushner.
Reports from earlier in the week say Kushner is a person of interest in that probe.
A staffer said White House officials are starting to realize the Russian Federation probe, and fallout over the administration's attempts to push back against the investigation, isn't going away.
Mr Kushner is understood to be under scrutiny by the FBI as part of its wider investigation into Russian links.
Kislyak, according to the Post, told Moscow that it was Kushner's idea to use the Russian diplomatic facilities in the us for this goal. The meeting was held in early December at Trump Tower in New York - Mr Trump's power base.
It is unclear why Kushner and Trump's team so direly needed a secret communications channel to the Kremlin, though given the near-daily attempts by Trump confidants to obscure or downplay their Russian connections, the answer may be unsavory.
American intelligence agencies first learned about the discussion several months ago, according to a senior American official who had been briefed on intelligence reports. The fact that they didn't want the content of those communications to be known even at a classified level can only mean that what they wanted to discuss is something they knew would be very damaging if they were made public.
This could be used to discuss Syria and other policy issues.
But neither side followed up on it.
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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.
The interactions between Mr. Trump's advisers and Mr. Kislyak have been a constant source of trouble for the new administration.
Mr Kushner's lawyer has previously said his client would co-operate with any inquiry.
The cascade of news showed that the multiple Russian Federation inquiries are advancing and that leaks continue to spout from the government, giving administration officials no respite even as they trot the globe.
Entous said sources had confirmed that USA intelligence intercepted Kislyak's reporting of the meeting.
Since all of these previously undisclosed meetings took place before Trump was sworn in, it could raise potential problems for Kushner.
"Mr. Kushner participated in thousands of calls in this time period", Kushner's attorney, Jamie Gorelick, said in response. "He has no recollection of the calls as described", she said.
"But we have not received such information", Ms. Gorelick added.
The contacts between Trump campaign associates and Russian officials during the presidential campaign coincided with what United States intelligence agencies concluded was a Kremlin effort through computer hacking, fake news and propaganda to boost Mr Trump's chances of winning the White House and damage his Democratic opponent, Hillary Clinton. Richard Burr, R-N.C., and Mark Warner, D-Va., the committee's two senior members.