(Reuters) -

The Muslim Brotherhood has rejected pleas from international envoys to "swallow the reality" that Mohamed Mursi will not return as Egypt's president, the Brotherhood spokesman said on Monday.

The envoys, trying to resolve a political crisis brought on by the army's overthrow of the Islamist Mursi a month ago, had visited jailed Brotherhood deputy leader Khairat El-Shater in the early hours of Monday.

But he cut the meeting short, saying they should be talking to Mursi, spokesman Gehad El-Haddad said. But people briefed on the meeting said it was long, thoughtful and in some moments intense but constructive and useful.

From the other side, a senior military source said the army and interim government would offer to free some Muslim Brotherhood members from jail, unfreeze its assets and give it three ministerial posts in a move to end the crisis.

A source involved in the diplomatic initiative said the releases from prison were expected within hours.

Several thousand Islamist supporters marched through downtown Cairo calling for Mursi's reinstatement and denouncing the army general who led his overthrow.

Marchers chanted "Mursi, Mursi" and "We are not terrorists", and waved picture of the ousted leader.

The protest showed tension is still running dangerously high in Egypt despite the mediation effort by the United States, the European Union, Qatar and the United Arab Emirates.

"Things should move soon, otherwise we shall miss this opportunity. This is all still incredibly fragile." said the source involved in the diplomatic initiative.

Mursi became Egypt's first freely-elected president in June 2012, 16 months after the overthrow of U.S.-backed strongman Hosni Mubarak, who had ruled for nearly 30 years.

But fears that he was trying to establish an Islamist autocracy coupled with a failure to ease economic hardships afflicting most of Egypt's 84 million people led to huge street demonstrations, triggering the army move.

Speaking about the talks in recent days, Haddad said the envoys "still carry the position that we should swallow the reality and accept that the military coup has happened and try to recover with minimum damage".

"We refuse to do so," Haddad told Reuters.

There was no agreement on how to start talks, he added.

The state news agency said earlier that diplomats, including U.S. Deputy Secretary of State William Burns and European Union envoy Bernardino Leon, had met Shater after midnight at the Tora Prison where he is being held south of Cairo.

Shater is seen as the political strategist of the group that propelled Mursi to office last year, and was arrested on charges of inciting violence after Mursi's downfall.

He told the envoys that only Mursi could "solve the mess" and the only solution was "full restoration of constitutional legitimacy and reversal of the coup", Haddad said.

"They invited him for discussions but he ended it abruptly...then he walked out of the room," Haddad said.

Mursi is being held at an undisclosed location and facing an investigation into accusations including murder. Most of the rest of the Brotherhood's leadership is also in custody.

In Washington, State Department spokeswoman Marie Harf said Burns had no plans to meet Mursi. She suggested no breakthroughs were imminent.

"There is clearly much more work to do. We have the goal of helping the Egyptians get back to a democratically elected, inclusive government," Harf told reporters.

In another development, a senior military source said the army and interim government would offer to free some Muslim Brotherhood members from jail if they had not been involved in violence.

It would allow the Brotherhood to reopen offices that had been shut after Mursi's overthrow and permit it to legalize its position to run in new elections. It would also unfreeze the group's assets and give it three ministerial posts.

"The initiative will be made so that we can end the crisis and have the Brotherhood end their sit-ins," the military source told Reuters.

A political source familiar with the proposal confirmed the details. The military source also said the government and the military had not yet agreed on Mursi's fate.

"We want the legal process to take its course while the government is seeking for all charges against him to be dropped," he said.

VIOLENCE ON HOLD

The diplomatic push has so far helped to hold off further bloodshed between Mursi's backers and the security forces.

An EU source in Brussels said the mediators were still trying to build confidence between the various sides and did not want to raise expectations.

"The real thing at this stage is to bring people together so they can actually meet and discuss these issues and for that you have to build up some trust and that can be done by very concrete measures, releasing people, dropping charges, not pressing charges, not moving into the squares, lowering the tension," the source said.

Thousands of Mursi supporters remain camped out in two Cairo sit-ins, which the government has declared a threat to national security and pledged to disperse. The interim government said on Sunday it would give mediation a chance but warned that time was limited.

Almost 300 people have been killed in political violence since Mursi's overthrow, including 80 shot dead by security forces in a single incident on July 27.

During Monday's march, protesters sprayed graffiti on walls and statues calling army chief General Abdel Fattah al-Sisi, who led Mursi's overthrow, a murderer and a traitor.

Security forces made no attempt to disperse a crowd estimated by reporters at several thousand strong.

"The military came and stole our country, they stole everything," said Mahmoud Isuafi, a businessman from the Nile Delta city of Mansoura. "I want democracy. Where is my vote? I can no longer elect my leader so I protest instead."

The military has laid out a plan that could see a new head of state elected in roughly nine months. The Brotherhood, which spent decades in the shadows before Mubarak's downfall, says it wants nothing to do with it.

However, diplomats say the Brotherhood knows Mursi will not return as president and wants a face-saving formula for him to step down that guarantees it a stake in the political future.

Two U.S. senators, Republicans Lindsey Graham and John McCain, were also due in Cairo at President Barack Obama's request to meet members of the new government and the opposition.

Before leaving on the mission, Graham said the Egyptian military must back out of politics quickly or risk a cut of the $1.5 billion in aid it receives from Washington each year.

(Additional reporting by Tom Finn, Shaimaa Fayed and Shadia Nasralla in CAIRO and Lesley Wroughton in WASHINGTON, Writing by Angus MacSwan, Editing by Michael Georgy and Michael Roddy)

By Yasmine Saleh and Angus MacSwan