* US, EU envoys meet foreign minister, pro-Mursi camp
* Interim government pledges safe exit for protesters
* Allies of deposed president say respect rallies against him
* Army chief appears to rule out run for presidency (Updates with ElBaradei, diplomats)
By Tom Perry and Matt Robinson
CAIRO, Aug 3 (Reuters) - Egypt's army-backed rulers and allies of its deposed Islamist president gave the first signs on Saturday of a readiness to compromise, pressed by Western envoys trying to head off more bloodshed.
Faced with the threat of a crackdown on supporters of the Muslim Brotherhood, diplomacy appeared to pick up pace, a month to the day since Egypt's army deposed President Mohamed Mursi and plunged the country into turmoil.
Recognising for the first time the strength of popular protest against his one-year rule, Mursi's allies said they respected the demands of millions who took to the streets before his overthrow.
A spokesman said the Mursi camp, which has refused to abandon weeks of sit-in protests until he is reinstated, wanted a solution that would "respect all popular desires".
They told envoys from the United States and the European Union that they rejected any role in a political settlement for army chief General Abdel Fattah al-Sisi, who led Mursi's ouster, and wanted the constitution he suspended to be restored.
"I respect and hold in regard the demands of the masses that went out on June 30, but I will not build on the military coup," spokesman Tarek El-Malt told Reuters, relaying what the pro-Mursi delegation had told the envoys.
Asked whether the delegation had insisted on Mursi's reinstatement as part of any political deal, Malt, a member of the Brotherhood-affiliated Wasat party, said that was a detail for future discussion.
But given that Mursi's opponents insist he should not be part of the political solution, Malt said that "Sisi must also not be in the political equation". He said the pro-Mursi camp was ready to talk with the National Salvation Front, a loose alliance of leftist and liberal parties represented in the interim government installed by the army.
RAGE 'THE EASIEST THING'
In an interview with the Washington Post, Sisi appeared to rule out running for president himself, despite his growing popularity among some of the 84 million-strong population.
"You just can't believe that there are people who don't aspire for authority," Sisi told the interviewer when asked if he would stand for president. Asked "Is that you?", he replied, "Yes." The Post said the interview was conducted on Thursday.
Egypt's military has laid out a "road map" to elections in about six months. It promises a return to civilian government, having brought down the first freely elected president after 60 years of rule by military men.
The Brotherhood, an Islamist movement that spent decades in the shadows before winning power in elections after the 2011 fall of autocrat Hosni Mubarak, had spurned the road map.
But its supporters, camped out at two sites in Cairo, face the threat of being violently dispersed by security forces who shot dead 80 of them a week ago. Almost 300 people have been killed in political violence since Mursi's overthrow, and much of the movement's leadership is in custody.
The deposed president is being held in a secret location, under investigation on a raft of charges including murder.
Diplomats say the Brotherhood and its political wing, the Freedom and Justice Party (FJP), understand that Mursi will not return as president, but they want a face-saving legal formula for him to step down.
"Those empowered to speak for the FJP understand that Mursi is not coming back. But they are maintaining that as a negotiating position," a Western diplomat said. Another diplomat said the Mursi bloc had shown flexibility in Saturday's talks.
Analysts say civilians in the new government are also trying to promote a political solution despite resistance from security services that want to take a hard line on the Brotherhood.
"Rage is the easiest thing. It is the easiest thing to say, 'Let's crush the Brotherhood'," interim Vice President Mohamed ElBaradei told Al-Hayat TV. "There is no solution in Egypt that can be based on exclusion. Salafists, Brotherhood, securalists, liberals, whoever, we are condemned to live together."
He said talks with outside envoys would focus on "specific steps" to reduce tensions. "We will discuss together how in the next 48 hours how we can halt the violence, reducing the number of protests."
'CONTACTS' WITH BROTHERHOOD
Stepping back from a threat to disperse the Brotherhood protests, the government said on Friday it would blockade the camps, but not storm them.
On Saturday, the Interior Ministry offered protesters a "safe exit" and political integration. Spokesman General Hany Abdel Latif said they had been "brainwashed".
U.S. Deputy Secretary of State William Burns and European Union envoy Bernadino Leon were leading the diplomatic push, meeting Mursi's allies, interim Foreign minister Nabil Fahmy and interim President Adli Mansour. The United Arab Emirates, which has given the new government $3 billion in support, sent Foreign Minister Sheikh Abdullah bin Zayed, who also met with Fahmy.
Fahmy told reporters there had been some contact with the Brotherhood.
"I wouldn't use the word negotiation. There have been contacts between different figures. There is no desire to use force if there is any other avenue that has any potential for success," he said.
The crisis in the Arab world's most populous country has posed a dilemma for the United States and other Western governments, which had advocated democracy following the overthrow of Mubarak but grew increasingly uncomfortable with Mursi's Islamist leanings.
Many Egyptians shared that concern, and frustration grew over Mursi's failure to solve social and economic problems.
The interim government gained the United States' approval on Thursday when Secretary of State John Kerry said the army had been "restoring democracy" when it toppled Mursi. Cairo remains central to U.S. policy in the Middle East, notably because of its peace agreement with Israel.
(Additional reporting by Michael Georgy, Tom Finn and Omar Fahmy in Cairo, Phil Stewart in Washington; Writing by Angus MacSwan and Matt Robinson; Editing by Robin Pomeroy and Peter Cooney)
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