Among the alternative ideas raised at the meeting last week were trying to identify and target a terrorist-linked group with ties to the Brotherhood that has not yet been designated or limiting any designation’s scope to the Egyptian branch, officials said.

A former general, Mr. el-Sisi helped lead a coup in 2013 that deposed Mohamed Morsi, Egypt’s first democratically elected president and a former Brotherhood leader, and has reimposed strongman rule on Egypt. The Egyptian government deemed the Brotherhood a terrorist organization several years ago as part of a brutal crackdown on its supporters, and Mr. el-Sisi repeatedly pressed the Obama administration to follow suit. But the Obama team refused, for both legal and policy reasons.

The Muslim Brotherhood was founded in 1928 in Egypt, and in the 1940s formed a secret, armed wing to fight against British colonial rule. It renounced violence in the 1960s and later embraced electoral democracy instead, although some offshoots and former members have engaged in terrorism.

The push for sanctions on the Brotherhood is the latest of several significant foreign policy decisions by Mr. Trump that appear to have been heavily influenced by talking to autocratic foreign leaders without first being fully vetted by career government professionals — such as his abrupt choice, since partly reversed, to swiftly pull American troops out of Syria.

The Trump administration had weighed whether to designate both the Muslim Brotherhood and an arm of Iran’s military, the Islamic Revolutionary Guards Corps, as terrorist organizations during its chaotic first weeks in 2017. But the ideas lapsed amid objections from career professionals and the fallout from other capricious early steps, like Mr. Trump’s ban on visitors from several predominantly Muslim countries.

But this spring, the administration abruptly pushed through the terrorist designation for the Revolutionary Guards. Mr. Pompeo, who has the most important voice in the debate besides Mr. Trump’s because the secretary of state controls the list of designated terrorist organizations, announced sanctions on the Iranian military arm on April 8, the day before Mr. el-Sisi visited the White House.

The move against the Iranian military was the first time the United States had interpreted its counterterrorism laws as permitting an entity of a nation-state government to count as terrorists.