Morsi's administration has defended his decree as an effort to speed up reforms and complete a democratic transformation. Leftists, liberals, socialists and others say it has exposed the autocratic impulses of a man once jailed by Mubarak.
Morsi's office said he would meet Egypt's highest judicial authority, the Supreme Judicial Council, on Monday, and the council hinted at compromise.
Morsi's decree should apply only to "sovereign matters", it said, suggesting it did not reject the declaration outright, and called on judges and prosecutors, some of whom began a strike on Sunday, to return to work.
Justice Minister Ahmed Mekky, speaking about the council statement, said: "I believe President Mohamed Morsi wants that."
The protesters are worried that Morsi's Muslim Brotherhood aims to dominate the post-Mubarak era after winning the first democratic parliamentary and presidential elections this year.
A deal with a judiciary dominated by Mubarak-era judges, which Morsi has pledged to reform, may not placate them.
Banners in Tahrir called for dissolving the assembly drawing up a constitution, an Islamist-dominated body Morsi made immune from legal challenge. Many liberals and others have walked out of the assembly saying their voices were not being heard.
Only once a constitution is written can a new parliamentary election be held. Until then, legislative and executive power remains in Morsi's hands, and Thursday's decree puts his decisions above judicial oversight.
One Muslim Brotherhood member was killed and 60 people were hurt on Sunday in an attack on the main office of the Brotherhood in the Egyptian Nile Delta town of Damanhour, the website of the Brotherhood's Freedom and Justice Party said.
The party's offices have also been attacked in other cities.