Al-Iraqiya radio said Sunni religious scholars declared Monday as the first day of the holiday as the new moon was sighted in the northern regions of Samarra and Mosul, and in Khales, west of Baghdad.
The announcement was greeted with bursts of gunshots in the air across Baghdad.
Al-Iraqiya also quoted a statement from the Supreme Council for the Islamic Revolution in Iraq, the main Shia organisation, inviting the faithful to watch for the new moon on Tuesday evening -- meaning they will celebrate Eid al Fitr on Wednesday.
Other Shia religious leaders had previously said the Eid would fall on Tuesday.
Libya too is celebrating Eid on Monday, the state television said. Eid day is decided in Libya according to astrological calculations.
Call for ceasefire
Meanwhile in Iraq, a Sunni leader called Monday for a week-long ceasefire to allow peaceful celebration of Eid al Fitr holidays.
"I call on the resistance groups to suspend operations this week so that Iraqis can live in peace, without the blast of explosives, bombs and shooting," said Adnan al-Dulaimi, head of Iraq's Sunni religious administration or Waqf.
"I ask also the occupation forces not to deal with Iraqis as
terrorists. We are peaceful, not evil," he said in an Eid sermon at Baghdad's Oum al-Qora mosque.
Dulaimi also urged the US-led coalition to free detainees and prisoners "apart from those proven to have carried out criminal acts."