In an unusual two-page editorial, Akhbar Al-Yom demanded that Saddam be tried in an international court.
The allegation is the latest salvo in the war of words that erupted between Iraq and Egypt last week when Egyptian President Hosni Mubarak accused Saddam of provoking the US-British airstrikes against Iraq on Dec 16-19.
Iraqi Deputy Prime Minister Tariq Aziz replied in an editorial in an Iraqi newspaper that there was not a single "fair and intelligent person in Egypt" who shared Mubarak's opinion.
Saturday's edition of Akhbar Al-Yom opened with a full front-page cartoon in colour of Saddam sitting on a pile of skulls. The caption said: "Who will avenge the 5,596 Egyptians killed by Saddam Hussein?"
Egypt had previously accused Iraq of killing Egyptians in 1988.
But this was the first time that the government, or the government press, had put a precise figure to the deaths.
Egyptians workers were said to have been victimised in the competition for jobs when thousands of Iraqi soldiers were demobilised at the end of the 1980-88 Iran-Iraq war.
Cairo newspaper reports at the time spoke of "scores" of Egyptians being beaten to death by Iraqi workers and plane loads of Egyptian workers being deported from Iraq. The number of Egyptians working in Iraq during the war reached 2 million.
In his scathing editorial, Akhbar Al-Yom's editor-in-chief, Ibrahim Saada, said he got the 5,596 figure from a "senior-level source." He did not elaborate.
The editorial said the Egyptians were beaten, tortured and had their bones broken and limbs amputated in Iraqi prisons.
Saada said that Egypt should send a statement to an international court for crimes of genocide in which "we accuse Iraqi President Saddam Hussein personally of being the leading, responsible killer in a series of crimes that lead to the death of 5,596 Egyptians without any reason."
"I ask (Egyptian) Prime Minister Kamal el-Ganzoury to instruct Foreign Minister Amr Moussa to send an urgent complaint to the United Nations against the Iraqi president," Saada wrote.
In Baghdad Saturday, Babil newspaper - which is owned by Saddam's eldest son - published a cartoon depicting Mubarak as a belly dancer performing to music played by Arab Gulf leaders, with US President Bill Clinton and Secretary of State Madeleine Albright applauding.