Morsi's Islamist backers have rejected negotiations with the military-backed government, leaving the most populous Arab nation in an uneasy limbo.
Still, the delay by the security forces gave the Sunni Muslim world's top religious institution more time to try to ease the political tensions with a new initiative.
Authorities also showed no signs of meeting key demands by Morsi's Muslim Brotherhood to release top Islamists who have been detained and face criminal investigations.
A judge ordered the deposed president, detained since he was overthrown July 3, to be held for 15 more days pending investigations of charges he conspired in 2011 with Palestinian militants, a judicial official said.
As news leaked that police were going to cordon off access to the sit-in sites early Monday, protesters took to the streets by the tens of thousands, and many made their way into the protest camps, whose populations include many women and children. Authorities said they wanted to "avoid bloodshed" and delayed taking any action.
The Anti-Coup Alliance, which works with the Brotherhood, said in a statement that the swift response of the people to come to the main sit-in site at the Rabaah al-Adawiya Mosque "is a great message to all parties that deserves our utmost respect."
The group also urged police not to respond to orders to blockade the sit-ins.
"Their rifles and bullets must only target enemies of Egypt," the group said.
For weeks, the government has been warning protesters to disperse, describing the sit-ins as a security threat.
The Interior Ministry has depicted the encampments as a public danger, saying 11 bodies bearing signs of torture were found near both sites. Amnesty International has also reported that anti-Morsi protesters have been captured, beaten, subjected to electric shocks or stabbed. At least eight bodies have arrived at a morgue in Cairo bearing signs of torture, the human rights group said.
Reporters Without Borders said two journalists were beaten by Morsi supporters while covering a Brotherhood march Friday in Cairo. The group also criticized "harsh measures" taken by authorities against news media supportive of the Brotherhood, saying 52 journalists were arrested since Morsi's removal from office.
Both the protesters and the security forces blame each other for using live ammunition in two major clashes near the Rabaah encampment that have killed at least 130 Morsi supporters.
Further violence threatens not only to delay the transition to a democratically-elected leadership, but could also further weaken the economy after more than two years of political instability.
The protests – a main tool of expression after the closure of pro-Brotherhood TV channels – have also stopped traffic and cut off main roads, and are being used by the Morsi camp as a political tool to increase pressure on the interim leadership.
After night fell Monday, speakers at the Rabaah sit-in led the flag-waving crowd in chants of "The police are thugs!" and "Islamic law, not secular law!" Some in the throng hoisted children up on their shoulders as they cheered, waved and made V-for-victory signs.
Security officials in charge of riot police units said they had been given notice Sunday to prepare their forces to cordon off the Rabaah site and another protest across town near Cairo University in Giza. Reports emerged of units coming to Cairo from around the country to take part in the operation.
The security forces had planned to form cordons around the two sites as early as dawn Monday, allowing protesters to leave but preventing others from getting in, to minimize casualties before using water cannons and tear gas, officials told The Associated Press.
After thousands streamed in and swelled the size of the sit-ins, however, security officials became concerned about the increased chance of bloodshed, and they decided not to move on the camps.
"We were stunned by the masses" who came to the camps, one military official told the AP, speaking on condition of anonymity because he was not authorized to talk to the media. He added that a push into the sit-ins would trigger a "massacre."
U.S. State Department spokeswoman Marie Harf said Washington was "very deeply concerned today about the potential for violence in Egypt."
"From the start, we've emphasized to those within Egypt that violence only sets back the eventual cause that they claim to be working for and need to allow people to protest peacefully," she added.
While the delay in using security forces to end the sit-ins has helped ease tensions that had spiked overnight, Egypt remains on edge as to how the standoff will end.
Morsi was deposed by the military after massive demonstrations across the country June 30, demanding he step down over what protesters saw as his failure to govern inclusively and manage the economy. Many accused him of acting only on behalf of the Muslim Brotherhood.
Last month, hundreds of thousands rallied in the streets to answer a call by Egypt's military chief, Gen. Abdel-Fattah el-Sissi, to give him a mandate to act against "potential terrorism" by Morsi's supporters.
The Egyptian Initiative for Personal Rights warned Monday that associating terrorism with the Brotherhood allows the military and police to use full force to preserve national security, shifting the debate to stability rather than on human rights and democracy.
Meanwhile, nearly two weeks of efforts by the international community to end the standoff and find a peaceful resolution failed. A locally negotiated solution also faces obstacles.
Influential Brotherhood member Mohammed el-Beltagy said he turned down an offer by the head of Al-Azhar, the Sunni Muslim world's top religious institution, to negotiate a solution. El-Beltagy said top Al-Azhar cleric Ahmed el-Tayyib was not an impartial mediator because he backed the coup.
However, public figures close to the Brotherhood have been approached to take part in the Al-Azhar talks.
Another Brotherhood figure, Saad Emara, dismissed efforts to negotiate a solution, saying the group doesn't recognize the "initiatives from the post-coup era."
"The key to a resolution is the return of legitimate institutions, including the president," Emara said.
Harf, the State Department spokeswoman, said the U.S. was "continuing to engage with all parties to push towards an inclusive, democratic process."
She added that the U.S. was urging "an end to all politically motivated arrests and detentions and emphasize that this won't help Egypt move beyond the crisis."
Sectarian violence, meanwhile, has flared in southern Egypt, and insurgents have battled the military in the Sinai, with Morsi's removal appearing to have lifted the lid on Islamic militancy in various parts of the country.
"The country is at a standstill," said Abdel-Rahman al-Bagi, part of a group of anti-Morsi demonstrators camped in Cairo's Tahrir Square since his overthrow. "Nothing is functioning because of the Muslim Brotherhood."
The Brotherhood sought to use the momentum of support from those who joined the two pro-Morsi sit-ins Monday to organize more marches.
One protest disrupted traffic across a major bridge in Cairo. Another saw about 100 marchers in a metro station holding pictures of Morsi and demanding he be reinstated.
Protesters have been fortifying the sit-ins camps. In Rabaah, men with helmets, sticks and what appeared to be protective sports equipment guarded barricades made of sandbags, truck tires and bricks. They have also built three concrete waist-high barriers against armored vehicles.
Brotherhood spokesman Gehad el-Haddad says they have no plans to back down.
"If they disperse one, we create two," he said