CAIRO—Egypt's embattled presidency showed signs of ceding to some of its opponents' demands after tens of thousands of activists thronged around the presidential palace on Friday in a deepening of the country's constitutional crisis.

Muslim Brotherhood-backed President Mohammed Morsi would consider delaying a constitutional referendum scheduled for Dec. 15 if political opponents agree not to challenge the new date in court, his vice president, Mahmoud Mekki, said on television Friday evening.

In...