It is unclear whether Mr. Kushner would need to review highly classified information. His current portfolio — which includes acting as an intermediary with Mexico, trying to forge Israeli-Palestinian peace negotiations, participating in an economic dialogue with China and working on revising the North American Free Trade Agreement — seems unlikely to involve major intelligence or national security secrets. But Mr. Kushner, by dint of his relationship with Mr. Trump, has wide-ranging access to the president and the information that he sees, and senior advisers to the president typically require such access to perform their duties.

The fact that the White House chief of staff would take the step of publicly denying that a policy change would harm the president’s son-in-law pointed up the tension in the West Wing after the Porter episode, particularly between Mr. Kelly on one side, and Mr. Kushner and his wife, Ivanka Trump, who had been close allies of Mr. Porter, on the other.

Mr. Kushner and Ms. Trump have been critical of Mr. Kelly in conversations with the president, who spent the weekend at his Mar-a-Lago resort in Palm Beach, Fla., surveying people about whether he should fire his chief of staff. However, since Mr. Porter’s departure, one official said, Mr. Kushner and Ms. Trump have told people around the White House that they have been vocal in their attempts to defend Mr. Kelly but are being treated unfairly in return.

One person familiar with Mr. Kushner’s thinking, who insisted on anonymity to describe it, denied that he felt personally targeted by Mr. Kelly or was agitating to have him removed. Another White House official denied that Mr. Kushner had ever raised the issue of the intelligence summary in his discussions with Mr. Kelly over his clearance.

But the memo, deliberately or otherwise, has shone an unflattering light on Mr. Kushner, raising questions about whether he can be effective in his post and how much authority he has. That debate threatens to complicate what Mr. Kelly has acknowledged is a long-overdue effort inside the White House to get a handle on the clearance process, a national security imperative over which top officials appear to have placed little priority after Mr. Trump took office.

“We should — and in the future, must — do better,” Mr. Kelly said in his memo last week.

The questions surrounding Mr. Kushner’s clearance are particularly acute because of the possibility that his extensive contacts with foreign actors — including travel, meetings with leaders overseas and multiple business ventures — might be relevant to the investigation by the special counsel, Robert S. Mueller III, into Russian meddling in the 2016 presidential election.

Mr. Kushner initially failed to disclose scores of those contacts on the standard form required for all prospective government officials, and has since amended his submission, substantially delaying his background check.