Kushner is Trump's son-in-law and holds the title of Senior White House Adviser.
The Washington Post and other major outlets reported last week that Kushner and Russian officials discussed the possibility of opening up a secret channel for communications a year ago.
Trump spent Sunday meeting with attorneys at the White House, presumably to set strategy for dealing with the Russian Federation investigation that has largely sidetracked his legislative agenda.
Trump also wrote that "many of the leaks coming out of the White House are fabricated lies".
And, there is no sign that Trump has sidelined him, as the president told The New York Times, "Jared is doing a great job for the country".
To drive home the size of these expectations, Trump turned his attention directly to Kushner - the grandson of Holocaust survivors, who still observes the Sabbath - to say: "If you can't produce peace in the Middle East, nobody can, okay?"
The Associated Press and other news organizations reported that Kushner in December proposed a back channel between the Kremlin and the Trump transition team.More news: Judi Dench loves India, says Ali Fazal
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Spicer, in his first White House briefing since the president returned from his foreign trip, refused to comment on something that is "not a confirmed action", despite Mr. Trump's retweet of the Fox article, which was based on an anonymous source.
"It's clear that Jared Kushner will be under intense scrutiny at a time when his father-in-law has named him everything but chief cook and bottle washer", Democratic strategist David Axelrod, a former top White House adviser to President Barack Obama, told Politico. Also present at the meeting was Trump's former national security adviser Michael Flynn, who is being investigated into whether he accepted money from foreign governments without the required approval.
Kushner's involvement in the proposed back channel was first reported by The Washington Post, which said he suggested using Russian diplomatic facilities for the discussions, apparently to make them more hard to monitor.
The FBI is reportedly looking into Kushner and the Kislyak conversation as part of its probe into Russia's hacking of the 2016 campaign and possible collusion with the Trump campaign.
The administration had hoped Trump's nine-day turn in the worldwide spotlight would offer a respite from relentless coverage-marked by an endless barrage of explosive revelations-of an ever-widening probe into Russian meddling in last year's election and possible collusion by the Trump campaign. He remains an influential confidant within the White House as does his wife, Ivanka Trump.