As Iraqi forces struck targets in and around the jihadist bastion, which saw deadly battles in 2004 between insurgents and American forces, IS claimed bombings in neighbouring Syria that killed at least 148 people.
The jihadist group has increasingly turned to its traditional tactic of killing civilians in bombings as it faces battlefield losses, and spokesman Abu Mohammed al-Adnani appeared to acknowledge in a recent statement that IS would probably lose more ground.
Abadi said the operation was supposed to start earlier, but "political problems and also the events... Threatening security inside Baghdad delayed some of the preparations".
Iraq has been hit by a months-long political crisis that has paralysed the legislature, and demonstrators have twice broken into the fortified Green Zone area, storming parliament and Abadi's office.
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IS has also carried out a series of deadly attacks in and around Baghdad this month.
Military sources said early efforts had isolated Fallujah from Karma, an area to the northwest where IS is still present.
IS, meanwhile, issued a statement claiming it had repelled "a wide attack" by Iraqi forces and destroyed multiple tanks and bulldozers.
Abadi's announcement settled the issue of which IS-held city Iraq should seek to retake next -- a subject of debate among Iraqi officials and international forces helping Baghdad battle the jihadists.
Iraq's Joint Operations Command yesterday warned civilians still in Fallujah -- estimated to number in the tens of thousands -- to leave.
It warned families that could not depart to raise a white flag over their location and stay away from IS headquarters and gatherings.