WASHINGTON — President Trump was only days into office in January 2017 when he first began considering a move with the potential to ignite a firestorm in the Middle East: designating the Muslim Brotherhood, a sprawling group that reaches from Morocco to Malaysia, a terrorist organization.

Mr. Trump’s hard-line national security adviser at the time, Michael T. Flynn, started a review of whether the United States should impose sweeping sanctions on millions of people across the Middle East. But only weeks later, after Mr. Flynn was fired, the proposal was quashed by Defense Secretary Jim Mattis, Secretary of State Rex W. Tillerson and Lt. Gen. H.R. McMaster, the next national security adviser.

Government lawyers had warned that the Muslim Brotherhood did not meet the legal criteria to be designated a terrorist organization. And in a volatile region where American troops were already battling Islamist extremists, the three men believed, taking on the Brotherhood was one fight too many.

Those men are now gone. So when President Abdel Fattah el-Sisi of Egypt, a strongman who views the Muslim Brotherhood as a source of political opposition, visited the White House on April 9 to renew his longstanding calls for Mr. Trump to target his foes, he found himself pushing on an open door.