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Ms. Azzam of Chatham House noted that Muslim Brotherhood leaders had called off a rally Tuesday, citing the need to “defuse tensions” between the fighting sides. She said this represents a willingness of the Brotherhood to keep the peace in support of the president.

“They don’t want any clashes between opponents,” she said.

A spokesman for the Brotherhood said that in some cities outside the capital, supporters were protecting Brotherhood offices. Morsi supporters said more than a dozen of their offices had been ransacked or set ablaze since Friday.

Some 5,000 demonstrated in the southern city of Assiut in support of Mr. Morsi’s decrees, according to witnesses there.

Protest organizers on a stage in the square called for another mass rally Friday. If the Brotherhood responds with mass rallies of its own, as some of its leaders have hinted, it would raise the prospect of greater violence after a series of clashes between the camps in recent days.

Opposition protesters warn of a situation that is likely to turn combustible if not resolved.

“If the current situation in Egypt continues, it will lead to more chaos — both the presidency and the Muslim Brotherhood have to understand how dangerous the situation is,” said Ahmed Maher, an organizer of the April 6 Youth Movement, a group known for its digital activism.

“We have smelled freedom, and it has made us forget fear,” he said. “We want to taste it now.”

Michael Scaturro and Louise Osborne in Berlin contributed to this article, which is based in part on wire service reports.