The army statement, read out on television Monday, had given Morsi 48 hours to comply with its call.
"If the demands of the people are not met in this period... (the armed forces) will announce a future roadmap and measures to oversee its implementation," it said.
But in a statement issued overnight, the presidency insisted it would continue on its own path towards national reconciliation.
The army declaration had not been cleared by the presidency and could cause confusion, it said.
The presidency also denounced any declaration that would "deepen division" and "threaten the social peace".
Morsi was consulting "with all national forces to secure the path of democratic change and the protection of the popular will", it added.
"The civil democratic Egyptian state is one of the most important achievements of the January 25 revolution," said the presidency, referring to the 2011 uprising that toppled dictator Hosni Mubarak.
"Egypt will absolutely not permit any step backward whatever the circumstances," it added.
The army's statement came just a day after millions of protesters took to the streets across Egypt Sunday, calling for Morsi to step down.
It received a rapturous welcome from Morsi's opponents, still camped out in Cairo's Tahrir Square.
Tamarrod, the grassroots campaign behind Sunday's massive protests against Morsi, also hailed the statement by the armed forces which it said had "sided with the people".
It "will mean early presidential elections", Tamarrod's spokesman Mahmud Badr told reporters.
Tens of thousands of jubilant protesters poured into the streets of Cairo, Alexandria and other large cities after the statement was broadcast. Cheers rang out across main squares.
In Tahrir, protesters voiced their support for army chief and Defence Minister General Abdel Fattah al-Sisi, chanting: "Come down Sisi, Morsi is not my president."
The army, which led a tumultuous transition after the 2011 revolution that ousted Mubarak, had given all parties one week to reconcile their differences.
The huge turnout for Sunday's protests, which the military put at millions nationwide, handed the initiative to the opposition Tamarrod movement.
Tamarrod had issued its own ultimatum, giving Morsi until 5:00 pm (1500 GMT) on Tuesday to quit or face an open-ended campaign of civil disobedience.
Sixteen people died in protests on Sunday, including eight in clashes between supporters and opponents of the president outside the Cairo headquarters of the Muslim Brotherhood, to which Morsi belongs, the health ministry said.
Early on Monday, protesters set the Brotherhood's headquarters ablaze before looting it.
Latest reports indicate that Egyptian Foreign Minister Kamel Amr has become the sixth minister to tender his resignation from the Morsi government on Monday.