This article from Reuters details a deal that was in place, brokered by the EU with the opposition, that was spurned by Morsi and the Muslim Brotherhood, two months before he was pushed from power:
Under a compromise crafted in months of shuttle diplomacy by EU envoy Bernardino Leon, six secular opposition parties would have recognized Mursi’s legitimacy and agreed to participate in parliamentary elections they had threatened to boycott.
In return, Mursi would have agreed to replace Prime Minister Hisham Kandil and five key ministers to form a technocratic national unity cabinet, sack a disputed prosecutor general and amend the election law to satisfy Egypt’s constitutional court.
The article does not mention the ‘five key ministers’, but the guess is that they were the Brotherhood appointees in charge of influential posts in education, information, supply, and the like. The former prosecutor general was fired and the new one appointed in a process contrary to the constitution Morsi swore to uphold.
But the negotiations didn’t work:
People familiar with the talks said Saad el-Katatni, leader of the Brotherhood’s political wing, helped negotiate the deal but could not sell it to Mursi and key Brotherhood leaders.
A very important caveat:
Mursi, Katatni and senior aides are detained by the army at unknown locations and cannot tell their side of the story.
However, right until the moment the military toppled him on July 3, the president went on proclaiming his electoral legitimacy and showed no signs of willingness to share power.
Of course by right he did not need to. But this report indicates opposition efforts to work with Morsi were not just cover for an eventual ‘coup’:
On that trip, Ashton also met armed forces commander General Abdel Fattah al-Sisi, the man who led the military intervention to oust Mursi. Participants said Sisi had also supported the EU initiative, saying the army did not want to intervene in politics and would welcome a broader national consensus.
“Contrary to what the Brotherhood is saying now, the army did its best to keep Mursi in office,” one participant said.
The full story is yet to be written, but if accurate this report provides important details from behind the scenes.