In one of the memos, White House aides were told that all materials prepared for the president must go first to Mr. Porter for vetting and clearance. Then Mr. Kelly must sign off on them before they go to Mr. Trump’s desk. That includes news articles, according to West Wing officials who described the memos’ content — of particular importance, given the propensity for some of Mr. Trump’s staff to slip him news accounts from dubious sources that shape his thinking or prompt him to cite unreliable or inaccurate information.

The process for documents with legal implications is somewhat more rigorous, although a version of it has been in place since Mr. Priebus was chief of staff.

For instance, the draft of an executive order will go through several stages of development, involving the White House Counsel’s Office and vetting by relevant staff and agency officials. Then it will go to Mr. Kelly for final approval before it is given to the president.

Mr. Priebus had tried to take firmer control of that process after Mr. Trump’s first week in office, when Stephen K. Bannon, the recently departed chief strategist, and Stephen Miller, the president’s senior policy adviser, pushed across the president’s desk two orders redesigning the National Security Council, and putting in place the first, court-contested travel ban on seven predominantly Muslim countries.

In the second memo, Mr. Kelly and Mr. Porter set up a system for deciding policy issues that have a legal aspect, such as executive orders, to ensure that all sides of the issue are heard. Any decision made in a meeting will be formalized in a memo that has to go through Mr. Porter and then Mr. Kelly for final sign-off.

That process is expected to curtail freelancing and hijacking of decisions by West Wing aides. Mr. Bannon was often accused of taking advantage of the loose process, but Mr. Trump’s son-in-law, Jared Kushner, and his daughter, Ivanka Trump, who both work in the West Wing, have also frustrated their colleagues for months by going directly to the president on specific issues.