CAIRO — Facing the most serious crisis of his presidency, Mohamed Morsi is leaning more closely than ever on his Islamist allies in the Muslim Brotherhood, betting on their political muscle to push through a decisive victory in the referendum on Egypt’s divisive draft constitution.

As tens of thousands chanted for his downfall or even imprisonment in a fourth day of protests outside the presidential palace, Mr. Morsi’s advisers and Brotherhood leaders acknowledged Friday that outside his core base of Islamist supporters he feels increasingly isolated in the political arena and even within his own government. The Brotherhood “is who he can depend on,” said one person close to Mr. Morsi, speaking on the condition of anonymity to discuss internal deliberations.

Mr. Morsi appears to believe that he and the Brotherhood can deliver a strong vote for the draft constitution in next Saturday’s referendum — strong enough to discredit the opposition, allow him a fresh start and restore some of his authority.

Struggling to quell protests and violence around the country, Mr. Morsi appeared to offer a new concession to his opponents Friday by opening the door to a possible delay in the referendum on the draft constitution, now scheduled for Dec. 15, and even potential revisions by the Islamist-dominated constitutional assembly.