The statement by Yasser al-Atta, a deputy commander of the army, comes after the army claimed advances in Omdurman, part of the wider capital, and an appeal by the United Nations Security Council for a truce during Ramadan, which begins this week.
The paramilitary Rapid Support Forces (RSF) said it welcomed the ceasefire call.
Atta's statement, issued on the army's official Telegram channel on Sunday and based on comments he made the previous day in Kassala state, said there could be no Ramadan ceasefire unless the RSF complied with a commitment made in May at Saudi and US-mediated talks in Jeddah to withdraw from civilian homes and public facilities.
It also said there should be no role for Mohamed Hamdan Dagalo, the RSF leader commonly known as Hemedti, or his family in Sudan's future politics or military.
"There are those who talk about a truce in Ramadan," Atta said in his Kassala speech to army graduates.
"There is no truce by order of the army and the people."
War between Sudan's army and the RSF erupted in mid-April 2023 amid tensions over a plan for transition to civilian rule.
The two factions staged a coup in 2021 that derailed a previous transition following the 2019 overthrow of autocratic former leader Omar al-Bashir.
A top Sudanese army general has ruled out any future role for RSF leader Mohammed Hamdan Dagalo. (AP PHOTO)
The army has been on the back foot militarily for much of the conflict, which has devastated swathes of the capital, triggered ethically driven killings in Darfur, and led to the world's biggest displacement crisis.
The RSF seized large parts of the capital in the first days of fighting.
However, the army has recently recovered some ground in Omdurman, which along with Khartoum and Bahri makes up the wider capital divided by the Nile.
One resident of Omdurman's Wad Nubawi district, Mohamed Abdel Rahman, said locals returning to the area taken by the army wanted to help restore services and repair the war damage visible all around in pockmarked and shattered buildings, destroyed shops and kiosks and burned-out vehicles.
"We want to work as youth with official bodies to bring back our families, and to hopefully start fasting for the month of Ramadan in our homes," he said, as heavily armed soldiers patrolled nearby.
The RSF said in a statement on Sunday that the army had rejected its offer to hand over 537 prisoners of war in its custody through the International Committee of the Red Cross.
"The SAF (Sudanese Armed Forces) leadership refused to accept our proposal to return their personnel as a sign of goodwill during Ramadan, as well as our previous unilateral initiatives since the current crisis began," the RSF said.
The army said it had received no communication from the ICRC about such an offer, and said the RSF's claims were "lies".
Several international attempts to broker ceasefires have failed to stop the fighting.
The UN says almost 25 million people - half Sudan's population - need aid, some eight million have fled their homes and hunger is rising.
Washington accuses both warring parties of committing war crimes.
UN Secretary-General Antonio Guterres has also called this week for a Ramadan truce.
Sudan's UN ambassador told the UN security council on Thursday that the head of the army and ruling council Abdel Fattah al-Burhan commended Guterres' appeal, but was wondering how it could be implemented.
Sudan's foreign ministry, which is aligned with the armed forces, said that to make any ceasefire possible, the RSF would need to withdraw from areas including El Gezira and Sennar states and several cities in Darfur, the RSF's stronghold.
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