AbdulMawgoud Dardery, a former member of parliament and a Muslim Brotherhood representative, told CNN's Amanpour program that the military could be an "honest broker" in a national dialogue. He said Morsy has reached out to opposition leaders many times, but the opposition "is afraid of democracy."
"It failed in the previous five elections we had in Egypt since the revolution, and they don't want to fail a sixth time," he said. "That's why they're going to street politics. Street politics is not an end in itself. It is a means to achieve democracy. But they're not willing to go toward a democratic system."
On Monday, protesters stormed the main headquarters of the Muslim Brotherhood, the party that Morsy led before his election. Armed with Molotov cocktails, the mob set the office on fire, shouting, "The people have toppled the regime."
At least 16 people were killed and more than 780 were wounded Sunday and Monday during the unrest in Egypt, the nation's health minister said, according to the official Egypt News agency.
On Friday, Andrew Pochter, a 21-year-old American in Alexandria to teach children English, was stabbed to death while watching the demonstrations, his family said.
Dr. Mohammed Mustafa Hamid told the news agency that eight people alone were killed in clashes at the Muslim Brotherhood's national headquarters in Cairo. All but 182 of the wounded have left the hospital after receiving treatment for their injuries.
State-funded Egyptian daily Al-Ahram also reported 46 sexual assaults during anti-Morsy protests in Egypt since Sunday, citing volunteer group Operation Anti-Sexual Harassment.
Opponents say Morsy's policies also are to blame for a breakdown in law and order and for a gas shortage that has Egyptians waiting at the pumps for hours. Monday's military statement seemed to adopt the protesters' perspective, calling the crisis a grave threat to national security while praising demonstrators as determined and admirable.
"Wasting more time will only lead to more division and fighting which we have and continue to warn against," the military said in its statement.
Morsy's supporters held smaller rallies Monday in other parts of Cairo. Some of them said he is the people's choice, that he inherited a broken system and should be given time to fix it.
"We're not leaving, and the president is staying," one supporter told CNN. "We believe in democracy. If people don't like him, they can vote him out in three years."
Dardery said Monday that the party might support early parliamentary elections. But he said the nation elected Morsy to a four-year term and should stand by that. To do otherwise would disrupt the country's nascent democracy, he said.
"It is not fair. It is not fair to a democracy," he said.
The developments were being closely watched around the region and in the United States, Egypt's leading ally. Speaking in Africa, U.S. President Barack Obama noted the protests and their demands.
"Our commitment to Egypt has never been around any particular individual or party. Our commitment has been to a process," he said.
Washington provided tens of billions of dollars in military and economic aid to Egypt under Mubarak and pledged $1 billion to the post-Mubarak government. Pentagon spokesman George Little told reporters Monday that the U.S. Defense Department is reviewing the latest statement from the Egyptian military, but no one is sure what will happen "one way or the other" in the next two days.