Hillary Clinton has called reports that six Trump White House officials used private email accounts for official business the "height of hypocrisy".
President Donald Trump repeatedly attacked Mrs Clinton's use of personal email while she was secretary of state during the 2016 campaign.
Mr Trump's son-in-law Jared Kushner admitted to using his email while five other aides reportedly did so as well.
Congress has asked the White House to provide details on the practice.
Speaking to Sirius XM radio on Monday, Mrs Clinton said: "It's just the height of hypocrisy."
She said the Trump administration "knew that there was no basis for all their hyperventilating" about her emails.
"It is something that if they were sincere about it, I think you'd have Republican members of Congress calling for an investigation," she added.
The former Democratic presidential candidate added that her use of a private server at the State Department was "a dumb mistake but a dumber scandal".
She said she regretted the country "had to go through it".Mr Kushner, a senior White House adviser, has been asked to preserve all his emails.
The New York Times has named the 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.
Meanwhile, Newsweek magazine says it has details of an email Ivanka Trump sent about collaboration with a business organisation, copying in two federal officials.
Who is behind the investigation?
The House Committee on Oversight and Government Reform, which has a responsibility for monitoring the effectiveness and accountability of federal government, has sent a letter to the White House demanding more information.
The letter was signed jointly by Republican Trey Gowdy, who chairs the committee, and Democrat Elija Cummings, the second most senior member.
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."
The letter sets a deadline of 9 October for the disclosure of more information.
Are private emails illegal?It is not illegal for White House officials to use personal email accounts for government business.
However, under the Presidential Records Act and Federal Records Act, government officials must forward any official correspondence to a work account within 20 days for preservation.
If this is not done reliably, the use of private accounts can put official records beyond the reach of journalists, lawmakers and others who seek publicly available information.
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".
There are also rules against sharing classified or privileged information on personal email accounts.
However, there is no suggestion that Mr Kushner or any of the others named did this.
Are Kushner and Clinton accused of the same thing?There are some key differences between the two cases.
Mrs Clinton had set up a private email server from her New York home - and used an email hosted by the server for all her emails, including at least 30,000 official emails, during her four years as secretary of state.
She did not use, or even activate, a state.gov email account, which would have been hosted on servers owned and managed by the US government.
The FBI said that "several thousand" work-related emails were not turned over to the State Department, and that Mrs Clinton had been "extremely careless" in handling classified information. However, an investigation into the matter was closed without charges.
It is not clear how many official emails Ms Trump, Mr Bannon, Mr Priebus, Mr Cohn and Mr Miller sent from their personal accounts.
However, White House spokeswoman Sarah Huckabee Sanders said that all White House employees had been told "if they receive work-related communication on personal accounts, they should be forwarded to official email accounts".