Aziz served as the country's foreign minister and, as its most senior Christian official, was the international face of the regime during its most tyrannical years.
Saad Yousif al-Muttalib, an advisor to Iraq's prime minister, was quoted in a television report as saying, "It will definitely take place, and it will take place after the Americans leave Iraq."
Aziz's execution would be highly controversial and could escalate political and sectarian tensions amid what critics say is an overly zealous campaign by Prime Minister Nouri Al-Maliki, a Shia Muslim, to crack down of the remaining elements of Saddam's Baathist supporters, most of whom are Sunnis.
But Al-Maliki's chief spokesman told Fox News that there has been "no decision to carry out the death sentence" due to disagreements between the president's office, which signs death warrants, and the prime minister's office.
"The warrant has not been signed because the presidency disagrees with some of the findings in the verdict against Aziz," Ali al-Dabbagh said. "The execution cannot happen until these legal complications are resolved."
He added: "The decision has nothing to do with the U.S. withdrawal, as he was sentenced to death some years ago."
Aziz, who is 75 and in poor health, surrendered to coalition troops shortly after the April 2003 invasion. He was eventually committed to 15 years in jail for crimes against humanity and then separately handed a death sentence by a top tribunal for "persecution of Islamic parties" -- one member of whom was Al-Maliki.
Al-Muttaib is one of the lesser known among al-Maliki's scores of advisers, and wider reaction to his comments have been largely received with surprise and skepticism.
Before Aziz's execution can take place, it still has to be broadly accepted among Iraq's highly factionalized politicians, who are split along deeply drawn sectarian and ethnic lines.
That acceptance, in Iraq's current environment, is a formidable task. It would be difficult for the Sunnis, as expected, to claim that by association his death was further evidence of al-Mailki's persecution of their bloc, but with escalating political violence underscoring the chasm of difference between the Shia-led government and its opponents, the intractable nature of some rivals is enough to spark further conflict.
While the level of violence is a tenth of what it was at the height of the war, months of rising sectarian tensions here have reached a new peak, with at least 28 people killed on Monday in attacks on pilgrims observing Ashura, the revered period of mourning for the Shia.