* Clashes break out in downtown Cairo
* Religious authority tries to convene talks
* No move yet against pro-Mursi protesters
* Crisis no nearer to solution after six weeks (Recasts with clashes, Brotherhood on talks)
By Tom Finn and Tom Perry
CAIRO, Aug 13 (Reuters) - Supporters and opponents of ousted Egyptian President Mohamed Mursi battled in downtown Cairo's streets on Tuesday, hurling rocks at each other as police fired volleys of tear gas.
The violence erupted as a proposal by al-Azhar, Egypt's leading religious authority, to bring together adversaries in the political crisis appeared to inch forward.
Mursi's Muslim Brotherhood said it was ready take part in talks as long as they were on the right terms.
But the clashes showed the country was still dangerously divided six weeks after the army overthrew Mursi.
Brotherhood protest camps at Cairo's al-Nahda Square and around Rabaa al-Adawiya mosque are the immediate focus of the crisis. Mursi supporters stood their ground behind barricades on Tuesday while Egypt's interim leaders debated how to end their sit-in.
No police crackdown appeared imminent despite frequent warnings from the army-installed government that the protesters should pack up and leave peacefully.
But clashes broke out in central Cairo when a few thousand Mursi supporters marched to the Interior Ministry.
Pro-army residents and shop-workers taunted them, calling them terrorists and saying they were not welcome. They then threw stones at the marchers, getting showered back in return.
Some hurled bottles at the Mursi supporters from balconies. Police then fired tear gas at the demonstrators. Women and children marchers fled the scene in panic.
The clashes spread to several streets and brought Cairo traffic to a standstill.
"There's no going forward with negotiations, the only way is back. Mursi must be reinstated," said Karim Ahmed, a student in a blue hard-hat who waved a picture of Mursi as he flung rocks at a ministry building.
Mursi took office in June 2012 as Egypt's first freely elected leader following the overthrow of long-ruling strongman Hosni Mubarak in a popular uprising the previous year.
But he failed to get to grips with a 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. Mursi and other Brotherhood leaders are now in jail.
Foreign mediators say the Brotherhood must accept that Mursi will not be reinstated. At the same time, the authorities must bring the Brotherhood back into the political process, they say.
The Brotherhood and the Salafi Nour party, Egypt's second largest Islamist party, both suggested on Tuesday they would be willing to join a meeting called by al-Azhar, whose initiative is the only known effort to end the crisis peacefully following the collapse of international mediation last week.
"If they stick to the rules we're asking for, yes," Brotherhood spokesman Gehad el-Haddad said, adding that talks must be based on the "restoration of constitutional legitimacy".
The Brotherhood would oppose proposals made by al-Azhar's Grand Sheikh Ahmed el-Tayeb, because he had supported the army's overthrow of Mursi, Haddad said. But he said there had been contacts with other al-Azhar officials.
Nour Party head Younes Makhyoun said his party had been invited to the al-Azhar talks.
"Currently the noble al-Azhar is trying to bring together for discussions those who have drawn up initiatives to agree, for example, on one initiative and vision, which we will use to pressure all the parties, so they accept it," he told Reuters.
The effort, however, was being complicated by the friction between the Brotherhood and al-Azhar, he said.
The government has its own plan for elections in nine months, but for now it is wrestling with the difficult issue of how to tackle the protest camps.
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 Brotherhood and want to move in.
More than 300 people have already died in political violence since Mursi's overthrow on July 3, including dozens of his supporters killed by security forces in two separate incidents.
The state-run al-Ahram newspaper reported that after a National Security Council meeting late on Monday, security forces were likely to cordon off the camps rather than take a more forceful approach that could lead to bloodshed.
A senior security source told al-Ahram that security had been bolstered around the camps to prevent weapons getting in.
Senior Brotherhood politician Farid Ismail said he did not expect the Interior Ministry to break up the encampments by force because of the likely casualty toll.
"It would be a big crime in addition to the crimes already committed, because it will result in a great cost in terms of massacres and dead," he said. "There are very large numbers, complete families, men, wives, children."
Also on Tuesday, a court set Sept. 7 as the start of another case brought against Mursi's allies, including prominent politician Mohamed el-Beltagi, on charges of kidnapping and torturing two members of the security forces. (Writing by Angus MacSwan in Cairo; Editing by Alistair Lyon)