John F. Kelly, who served as former President Donald J. Trump’s second White House chief of staff, said in a sworn statement that Mr. Trump had discussed having the Internal Revenue Service and other federal agencies investigate two F.B.I. officials involved in the investigation into his campaign’s ties to Russia.

Mr. Kelly said that his recollection of Mr. Trump’s comments to him was based on notes that he had taken at the time in 2018. Mr. Kelly provided copies of his notes to lawyers for one of the F.B.I. officials, who made the sworn statement public in a court filing.

“President Trump questioned whether investigations by the Internal Revenue Service or other federal agencies should be undertaken into Mr. Strzok and/or Ms. Page,” Mr. Kelly said in the statement. “I do not know of President Trump ordering such an investigation. It appeared, however, that he wanted to see Mr. Strzok and Ms. Page investigated.”

Mr. Kelly’s assertions were disclosed on Thursday in a statement that was filed in connection with lawsuits brought by Peter Strzok, who was the lead agent in the F.B.I.’s Russia investigation, and Lisa Page, a former lawyer in the bureau, against the Justice Department for violating their privacy rights when the Trump administration made public text messages between them.

The disclosures from Mr. Kelly, made under penalty of perjury, demonstrate the extent of Mr. Trump’s interest in harnessing the law enforcement and investigative powers of the federal government to target his perceived enemies. In the aftermath of Richard M. Nixon’s presidency, Congress made it illegal for a president to “directly or indirectly” order an I.R.S. investigation or audit.

The New York Times reported last July that two of Mr. Trump’s greatest perceived enemies — James B. Comey, whom he fired as F.B.I. director, and Mr. Comey’s deputy, Andrew G. McCabe — were the subject of the same type of highly unusual and invasive I.R.S. audit.

It is not known whether the I.R.S. investigated Mr. Strzok or Ms. Page. But Mr. Strzok became a subject in the investigation conducted by the special counsel John Durham into how the F.B.I. investigated Mr. Trump’s campaign. Neither Mr. Strzok nor Ms. Page were charged in connection with that investigation, which former law enforcement officials and Democrats have criticized as an effort to carry out Mr. Trump’s vendetta against the bureau. Mr. Strzok is also suing the department for wrongful termination.

Mr. Strzok and Ms. Page exchanged text messages that were critical of Mr. Trump and were later made public by Rod J. Rosenstein, then the deputy attorney general under Mr. Trump, as he faced heavy criticism from Republicans on Capitol Hill who were trying to find ways to undermine him.

The sworn statements from Mr. Kelly are similar to ones he made to The New York Times in November, in which he said that Mr. Trump had told him that he wanted a number of his perceived political enemies to be investigated by the I.R.S., including Mr. Comey, Mr. McCabe, Mr. Strzok and Ms. Page.

Mr. Kelly told The Times last year that Mr. Trump’s demands were part of a broader pattern of attempts to use the Justice Department and his authority as president against people who had been critical of him, including seeking to revoke the security clearances of former top intelligence officials.

In the sworn statement, Mr. Kelly said that Mr. Trump had discussed having the security clearances of Mr. Strzok and Ms. Page revoked, although Mr. Kelly did not take action on the idea. Mr. Kelly said that his notes showed that Mr. Trump discussed the investigations of the two on Feb. 21, 2018.

“I did not make a note of every instance in which then President Trump made a comment about Mr. Strzok and Ms. Page,” Mr. Kelly said. “President Trump generally disapproved of note-taking in meetings. He expressed concern that the notes might later be used against him.”

Mr. Kelly said that he never took any steps to follow through on Mr. Trump’s desires to have his enemies investigated.

Mr. Trump has said he knew nothing about the audits of Mr. Comey and Mr. McCabe and their spouses. The I.R.S.’s inspector general found last year that Mr. Comey and Mr. McCabe had been randomly selected for the audits, though the inspector general’s report acknowledged some deviations from the I.R.S.’s rigorous rules for random selection when the agency made final selections of the returns that would be audited.

Mr. Kelly told The Times last year that Mr. Trump had at times discussed using the I.R.S. and the Justice Department to address others in addition to Mr. Comey, Mr. McCabe, Mr. Strzok and Ms. Page.

They included, Mr. Kelly said, the former C.I.A. director John O. Brennan; Hillary Clinton; and Jeff Bezos, the founder of Amazon and the owner of The Washington Post, whose coverage often angered Mr. Trump.