Private emails: House committee ups pressure on White House
Mr Kushner, a senior adviser, has been asked to preserve all his emails.
His wife, Mr Trump's daughter Ivanka, is also said to have a private account.
The demand for more information came in a letter from the House Committee on Oversight and Government Reform, signed by Republican Trey Gowdy and Democrat Elija Cummings.
Addressed to White House Counsel Donald McGahn, it says: "Have you or any non-career official at the White House ever used a personal email account to conduct official business?
"If so, please identify the individual and the account used, and provide evidence of measures to ensure compliance with federal law," it reads.
The letter sets a deadline of 9 October for the disclosure of more information.
Mr Kushner's lawyer says that "fewer than 100 emails" were sent through a private account.
The New York Times has named the four other staffers implicated as Steve Bannon, the former chief White House strategist; Reince Priebus, the former chief of staff; and advisers Gary D Cohn and Stephen Miller.
Newsweek magazine says it has details of an email Ivanka sent about collaboration with a business organisation, copying in two federal officials.
Was it illegal?
There is no suggestion that Mr Kushner or any of the others named shared classified or privileged information via private email accounts.
It is not illegal for White House officials to use private email, as long as they forward professional messages to their work accounts for preservation.
Federal regulations specify how records related to the president and other government activities should be maintained.
If this is not done reliably, the use of private accounts can put official records beyond the reach of journalists, lawma
kers and others who seek publicly available information.
The situation also leaves the Trump family open to claims of hypocrisy, as President Trump has repeatedly criticised Hillary Clinton for using a personal email account while she was secretary of state.
On the campaign trail, he vowed to imprison his Democrat rival over her handling of classified information.
'The public has a right to certainty'
Mr Cummings, the ranking member of the House committee, sent a letter to Mr Kushner on Monday, quoting from the Republican investigation of Hillary Clinton's email server to justify asking him to keep his emails.
"The public has a right to access public records," he wrote, quoting Trey Gowdy's letter to Mrs Clinton's legal team on 19 March 2015.
"The public has a right to certainty that no classified or sensitive information was placed at risk of compromise.
"Your actions in response to the preservation request and the information you provide in response to this letter will help determine the next steps in this investigation," the Maryland congressman wrote to Mr Kushner, a former real estate investor.
In a statement Mr Kushner's lawyer said: "Fewer than 100 emails from January through August were either sent to or returned by Mr Kushner to colleagues in the White House from his personal email account."
He said most were news articles or political commentary and "all have been preserved in any event".
White House spokeswoman Sarah Huckabee Sanders would not commit to releasing the emails at Monday's press briefing, saying: "I'm not going to get ahead of a conversation that hasn't taken place."
She added that the use of private emails to conduct government business is "to my knowledge, very limited".