Mr. Kushner had also filled out paperwork for his White House security clearance wrongly and had to refile it repeatedly, CBS reported.
Kushner's responsibilities range from serving the president on such issues as China, Syria, Afghanistan and the Middle East, Politico reports, along with infrastructure and innovation.
US President Donald Trump's son-in-law and senior adviser voted in Novembers US presidential election as a woman, the media reported on Thursday.
"Here's a good reason why it matters that Kushner is using private emails-it's reportedly a "safe bet" that foreign governments hacked him", legal expert Renato Mariotti wrote on Twitter.
"The public has a right to certainty that no classified or sensitive information was placed at risk of compromise".
The paper has named the four other staffers as Steve Bannon, the former chief White House strategist; Reince Priebus, the former chief of staff; and advisers Gary D Cohn and Stephen Miller.
Gee, who would want to allow their personal cellphone to be used as a clandestine listening device for the Russians, as the NSA warned White House officials could easily happen.
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The site based its report on "one former senior USA intelligence official familiar with the briefings", and "and three other former officials familiar with such briefings".
For people who rode into the White House on disingenuous "lock her up" chants, it's an incredible irony that they appear to be guilty of something so much worse than simply using private email addresses and devices.
The sessions are held for each incoming administration, according to Politico.
A White House official reacted to the Journal's story by telling the newspaper that McGahn "did not consider resigning, and he was not concerned about any one individual".
"Kushner and others of his rank were expected to attend, especially given the high potential for cyberespionage as one administration replaces another", Politico reports.
Abbe Lowell, Kushner's laywer, indicated that "fewer than 100 emails about government business are in Kushner's private email domain", and they have been forwarded to the White House account, Politico added. "He was focused on implementing the proper processes and structures to protect the White House and its staff, including Jared".
"To my knowledge, it's very limited", she said.