The Muslim Brotherhood said it would not hold the protest on Tuesday, as originally planned, “to avoid clashes”.
Opponents of President Mursi and of the brotherhood have said they would hold their own protest against a decree giving the president sweeping powers.
Mr Mursi has been meeting senior judges in an effort to defuse the crisis.
Ahead of Monday’s meeting with members of the Supreme Judicial Council, he expressed confidence that a solution would be found.
“President Mursi is very optimistic that Egyptians will overcome this challenge as they have overcome other challenges,” spokesman Yasser Ali told reporters,
Later Mr Ali said the president had reassured the judges that the scope of the decree would be confined to key matters of “sovereignty”.
There is no word so far from the judges.
Several prominent opposition leaders, including Nobel Peace Prize laureate Mohamed ElBaradei, have said they will not engage in dialogue with the president until he rescinds the measure, known as the constitutional declaration.
The decree, issued last week, says no authority can revoke presidential decisions.
There is a bar on judges dissolving the assembly drawing up a new constitution. The president is also authorised to take any measures to preserve the revolution, national unity or safeguard national security.
The decree has sparked violent protests in Cairo and across the country.
On Sunday teenager Islam Fathy Massoud died and 60 people were injured in clashes in the Nile Delta town of Damanhour between the president’s supporters and opponents.
His funeral was held on Monday, while in Cairo thousands of people marched through Tahrir Square for the funeral of another young activist killed in recent clashes with police.
Egypt‘s stock market, which had seen a fall of almost 10% on Sunday, recovered some ground on Monday morning.