The former White House chief of staff, John F. Kelly, on Wednesday declined to answer questions about the existence of a memo he wrote saying that President Trump had ordered officials to give his son-in-law, Jared Kushner, a security clearance in May 2018.

Mr. Kelly also broke with Mr. Trump on key aspects of his approach to immigration and the NATO alliance, and said that his top concern about decisions made by the president was whether they were objectively right for the country when divorced from political concerns.

Mr. Kelly, who kept his voice level during a 90-minute question-and-answer session at Duke University, would not specifically address Mr. Kushner’s clearance being ordered by Mr. Trump, which The New York Times reported last week.

“I couldn’t — and I’m not dodging — I couldn’t comment on that for a couple of reasons,” Mr. Kelly said, citing clearances being among the things that he could not discuss, and that conversations with the president “at that level would certainly” be kept confidential under executive privilege.