The army would offer a “road map” for peace if Islamist President Mohammed Morsi and his opponents failed to heed “the will of the people”, it said.
It later issued a clarifying statement denying its warning amounted to a coup.
Given the inability of politicians from all sides to agree until now, it seems unlikely Mr Morsi can survive in power, says the BBC’s Aleem Maqbool in Cairo.
On Sunday millions rallied in cities nationwide, urging the president to quit.
Large protests continued on Monday, and eight people died as activists stormed and ransacked the Cairo headquarters of the Muslim Brotherhood, to which the president belongs.
He became Egypt’s first Islamist president on 30 June 2012, after winning an election considered free and fair following the 2011 revolution that toppled Hosni Mubarak.
The head of the armed forces described Sunday’s protests as an “unprecedented” expression of the popular will.
The statement by the minister of defence and army chief, Gen al-Sisi, was worded carefully.
It did not say the president must go. The army, with troops in strategic positions across Cairo, is saying the government and opposition have 48 hours to agree a way forward or it will intervene with its own plan.
The Egyptian military has been both hero and villain for the people involved in the ousting of President Mubarak in 2011.
Heroes, first of all, when they put themselves between protesters and the Mubarak regime’s enforcers. But later they were widely criticised for holding onto power for too long.
The reality is they have never given up their critical role behind the scenes, which includes huge economic power.
No matter which way Egypt goes – and there could be some very rough days ahead – the army will never want its own power diluted.
In a statement read out by a spokesman on state television on Monday evening, Gen Abdel Fattah al-Sisi said the army would not get involved in politics or government.
There were scenes of flag-waving jubilation in Cairo’s Tahrir Square, where supporters of Tamarod (Rebel) – the opposition movement behind the protests – interpreted the statement as spelling the end for a president they accuse of putting the Brotherhood’s interests ahead of the country’s as a whole.
As five helicopters flew over the square with huge Egyptian flags hanging below them, the crowds chanted: “The army and the people are one hand.”
In the Mediterranean city of Port Said, crowds set off fireworks and sang the national anthem in Martyrs’ Square. Some protesters clambered onto police vehicles to celebrate, in what BBC Arabic’s Attia Nabil at the scene says was a show of better relations between the police and the people.
And protesters conducting a sit-in outside Mr Morsi’s house in Zagazig pledged to remain until a clear plan for handing over power was enacted.
But a second statement posted on the military’s Facebook page late on Monday emphasised the army “does not aspire to rule and will not overstep its prescribed role”.
“Our earlier statement’s purpose was to push all parties to find a quick solution to the current crisis… to push towards a national consensus that responds to the people’s demands,” said the statement.
An undated photograph posted on the president’s official Facebook page showed Mr Morsi smiling with Gen Sisi and Prime Minister Hisham Qandil.
A senior member of the Brotherhood’s Freedom and Justice Party (FJP) said a solution to the crisis “will be in the framework of the constitution”.
“The age of military coups is over,” Yasser Hamza, a member of the FJP’s legal committee, told Al Jazeera TV.
And senior Brotherhood figure Muhammad al-Biltaji urged thousands of pro-Morsi supporters, gathered outside the Rab’ah al-Adawiyah mosque in Cairo’s Nasr district, to “call their families in all Egyptian governorates and villages to be prepared to take to the streets and fill squares” to support their president.
“Any coup of any sort will only pass over our dead bodies,” he said to a roar from the crowd.
Local media reported that a late-night press conference by the Egyptian presidency had been cancelled.
The opposition movement had given Mr Morsi until Tuesday afternoon to step down and call fresh presidential elections, or else face a campaign of civil disobedience.
On Saturday, the group said it had collected more than 22 million signatures – more than a quarter of Egypt’s population – in support.
But Mr Morsi was defiant in an interview published on Sunday, rejecting calls for early presidential elections.
Muslim Brotherhood spokesman Gehad Haddad told the BBC the roadmap referred to by Gen Sisi did not necessarily increase pressure on the president to call early presidential elections.
Rather, he said, the pressure was on Egypt’s constitutional court to swiftly issue a new parliamentary law and to call for parliamentary elections.
Meanwhile, the al-Watan website said the ministers of tourism, environment, communication and legal affairs had resigned in an act of “solidarity with the people’s demand to overthrow the regime”.
US President Barack Obama has called for restraint on all sides, saying the potential for violence remained.
Although it was not the job of the US to choose Egypt’s leaders, it wanted to make sure all voices were heard, said Mr Obama during a visit to Tanzania.