The state-run al-Ahram newspaper reported on Tuesday that the leaders are discussing a plan, signalling no imminent police crackdown on the sit-ins.
After a National Security Council meeting late on Monday, al-Ahram cited presidential sources as saying security forces were likely to cordon off the Islamist protest camps rather than take a more forceful approach that could lead to bloodshed.
The two main camps at Cairo's al-Nahda Square and around the Rabaa al-Adawiya mosque have become the focus of the political crisis since Morsi's overthrow by the military on July 3.
Thousands of his supporters have gathered there, denouncing the toppling of Egypt's first freely elected leader and saying they will stay until he is freed from detention and reinstated.
That poses a problem for the military-installed government as it presses on with its own plan for elections in nine months.
Some officials wish to avoid a bloody showdown, which would damage the government's efforts to present itself as legitimate, while hardliners in the army and security forces fear they are losing face to the Muslim Brotherhood and want to move in.
"The consultations are continuing among all government bodies. The most probable path is to encircle the two sit-ins, to choke them instead of [launching] a security intervention that could cause casualties," al-Ahram reported.
International efforts to resolve the crisis collapsed last week. Foreign mediators say Morsi's Muslim Brotherhood must accept that he will not be returned to power.
At the same time, the new authorities must bring the Brotherhood back into the political process, they say.
Morsi took office in June 2012 after elections that followed the overthrow of long-ruling strongman Hosni Mubarak in a popular uprising the previous year.
But he failed to get to grips with the country's deep economic malaise and worried many Egyptians with apparent efforts to tighten Islamist rule of the most populous Arab nation. The army ousted him amid huge demonstrations against his rule. Morsi and other Brotherhood leaders are now in jail.
Tarek el-Malt, a member of the Brotherhood-allied Wasat Party, said he had written to the interior minister to warn that using force against the protest camps would make matters worse.
Asked why the police had yet to move against them, he told Reuters: "The only interpretation is that, praise God, there are people who are still not thinking in an exclusionary fashion and want to give an opportunity for peaceful solutions."
Malt was part of a delegation of Morsi allies that met international envoys during their mediation attempt. He said he and his colleagues were ready to talk to interim vice-president Mohamed ElBaradei and Deputy Prime Minister Ziad Bahaa Eldin "as political symbols" to seek a way out of the impasse.
"We welcome dialogue with politicians in general, and even those in government, but not in their official capacities," he said. – Reuters