President Trump is being pressed for details on his daughter’s email use in office by the same Republican responsible for spearheading the congressional investigation into his former White House rival, Hillary Clinton, that uncovered details about her own email habits while secretary of state.

Rep. Trey Gowdy, the outgoing head of the House Oversight Committee, wrote the president’s chief of staff this week requesting further details about Ivanka Trump’s use of a private email address while serving as a formal adviser to her father.

“In light of the importance and necessity of preserving the public record and doing so in a manner that is reflective of relevant statutory and regulatory requirements, the Committee must assess whether the White House took adequate steps to archive Ms. Trump’s emails and prevent a recurrence,” Mr. Gowdy, South Carolina Republican, wrote in a letter Wednesday to John Kelly.

Mr. Gowdy, 54, led the House panel’s investigation into the 2012 terrorist attack at the U.S. consulate in Benghazi, Libya, taking particular interest in Mrs. Clinton’s handling of the incident as secretary of state under former President Barack Obama. The probe ultimately uncovered substantial details about Mrs. Clinton’s use of a private email server for government business, and Mr. Gowdy, a former federal prosecutor, later claimed that she should have been subsequently charged for mishandling classified information.

More recently, The Washington Post revealed on Monday this week that Ms. Trump, the president’s eldest daughter and an unpaid senior White House adviser, similarly used a private email account in 2017 to send hundreds of messages to various White House aides, administration officials and assistants while in office.

“Ms. Trump’s use of a personal email account for official communications may implicate the Presidential Records Act and other security and recordkeeping requirements,” Mr. Gowdy wrote in the letter to the White House this week.

Mr. Gowdy has asked the White House to provide a response by Dec. 5.

The White House did not immediately return a message seeking comment.

The president downplayed reports about his daughter’s emails when questioned about her conduct earlier in the week, contrasting her habits with his former White House rival.

“They aren’t classified like Hillary Clinton. They weren’t deleted like Hillary Clinton,” said Mr. Trump. “What Ivanka did, it’s all in the presidential records. Everything is there.”

“You’re talking about a whole different — you’re talking about all fake news … there was no nothing. What it is, is a false story,” Mr. Trump insisted.

Mr. Gowdy chose not to run for re-election during the Nov. 6 midterm elections, and his term in Congress is set to expire on Jan. 4, 2019, at which point Democrats are slated to take control of the House of Representatives.

“We launched a bipartisan investigation last year into White House officials’ use of private email accounts for official business, but the White House never gave us the information we requested,” said Maryland Rep. Elijah Cummings, a Democrat and the likely incoming chairman of the House Oversight Committee.

“We need those documents to ensure that Ivanka Trump, Jared Kushner and other officials are complying with federal records laws and there is a complete record of the activities of this Administration,” he said in a statement Tuesday.

Ms. Trump, 37, married Mr. Kushner, a New York businessman, in 2009. Both hold positions in the president’s administration, notwithstanding complaints of nepotism raised by his critics.

The FBI investigated Mrs. Clinton’s email use while secretary of state prior to ultimately deciding that her conduct was careless but not criminal — an assessment disputed by Republicans including both. Mr. Gowdy and Mr. Trump, among others.

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