CAIRO, Egypt -- Egypt's highest security body warned Sunday that the clock is ticking for a peaceful end to the standoff over sit-ins by supporters of ousted President Mohammed Morsi, suggesting that authorities will break up the vigils unless mediation efforts produce results soon.

More than a month after the military overthrew Morsi, tens of thousands of the Islamist leader's supporters remain camped out in two main crossroads in Cairo demanding his reinstatement. Egypt's military-backed interim leadership has issued a string of warnings for them to disperse or security forces will move in, setting the stage for a potential bloody showdown.

Also Sunday, authorities announced a court case accusing the spiritual leader of the Muslim Brotherhood and his powerful deputy of inciting murder will start Aug. 25. Morsi hails from the Brotherhood.

The U.S. and EU are trying to find a peaceful resolution to the standoff to avoid a repeat of violence that has killed more than 250 people -- at least 130 of whom pro-Morsi protesters shot dead by security forces in two clashes -- since the July 3 military coup.

While diplomats raced to find a compromise, the Egyptian interim government signalled that its patience with the pro-Morsi sit-ins was running out.

The National Defence Council, which is led by the interim president and includes top Cabinet ministers, said the search for a peaceful resolution is not open-ended. The council said a negotiated resolution also would not shield from legal proceedings what it called "law-breakers" and others who incite against the state.

It said a chance should be given to all "negotiations and mediations" that could end the protests without bloodshed, but that the timeframe should be "defined and limited." It also called on the protesters to abandon the sit-ins and join the political road map announced the day of Morsi's ouster.

With the Islamist-backed constitution adopted last year suspended and the legislature dominated by Morsi's supporters dissolved, the road map provides for a new or an amended constitution to be put to a national referendum later this year and presidential and parliamentary elections early in 2014

In a move that underlined the government's resolve in dealing with the protests -- now in their second month -- Egyptian authorities on Sunday denied Yemen's Nobel Peace Prize winner Tawakkul Karman entry into Egypt after she landed at Cairo airport on Sunday.

Karman, the first Arab woman to win the Nobel Peace prize, has stated her opposition to Egypt's military coup and said she had intended to join the pro-Morsi sit-in protests.

Airport officials said she was sent back on the Sunday flight that brought her to Cairo from the United Arab Emirates. They did not say why she was denied entry, only that her name had been placed by various security agencies on a stop list at the airport. The officials spoke on condition of anonymity because they were not authorized to brief the media.

The decision to bar Karman suggests authorities wanted to deny the pro-Morsi camp the publicity she would have generated and the idea that prominent figures outside Egypt also oppose Morsi's ouster. Morsi supporters strongly condemned Karman's barring, claiming it was evidence of the "resurrection" of the police state Egypt had under autocrat Hosni Mubarak, toppled in a 2011 popular uprising.

Karman shared the Nobel Peace prize in 2011 with Liberia's President Ellen Johnson Sirleaf and women's rights campaigner Leymah Gbowee. She earned it for her role in the protests that swept Yemen in 2011 to force longtime dictator Ali Abdullah Saleh from office.

Pro-Morsi protesters blocked a major road Sunday that runs through most of the city and leads to its international airport.

Meanwhile, U.S. Deputy Secretary of State William Burns extended his visit to Cairo by one day so he can have further talks with Egyptian leaders. He met Defence Minister Gen. Abdel-Fatah el-Sissi, who led the coup, on Sunday. A member of the pro-Morsi delegation that met Saturday with Burns said the four delegates also would hold another round of talks with the U.S. diplomat.

Other top diplomats in Cairo are the foreign ministers of the United Arab Emirates, which had been at sharp odds with Morsi's government, and Qatar, which maintains close ties to the Brotherhood. European Union's special envoy, Bernardino Leon, is also in Cairo.

At the core of discussions is the political future of the Muslim Brotherhood and its Islamist allies. The Brotherhood says it is looking for concessions before beginning talks with the military-backed administration. These measures could include releasing detained Brotherhood leaders, unfreezing the group's assets, lifting a ban on Islamist television stations loyal to Morsi and reigning in the use of force against its protesters.

Morsi has been held at undisclosed locations since July 3. He faces accusations of comprising with the militant Palestinian Hamas group to escape prison in 2011. Morsi has been visited by Catherine Ashton, the EU's foreign policy chief, and a delegation of African statesmen. Ashton reported that he was well and had access to television and newspapers.

Egypt's state news agency said on Sunday that Brotherhood leader Mohammed Badie and his deputy Khairat el-Shater are to stand trial Aug. 25 for complicity and incitement in the killing of eight demonstrators outside the group's Cairo headquarters.

Badie is at still large, while el-Shater is in custody.

The killings took place during the first day of the mass street protests calling for Morsi's ouster. The agency also said that senior Brotherhood figure Rashad Bayoumi will face trial on the same charges. Three others face murder charges in the same case.

Morsi's palace aides Rifaah el-Tahtawi and Asaad el-Sheikha meanwhile faced questions over allegations they illegally held and tortured anti-Morsi protesters last Dec. 5 after supporters of the ousted leader descended upon a group of unarmed demonstrators camped outside the presidential palace. Clashes lasting all day left at least 10 dead and hundreds injured.

Both el-Tahtawi and el-Sheikha are in detention.