Lieutenant General Abdel Wahab al-Saadi said his forces secured the largely agricultural southern neighbourhood of Naymiya, under cover of US-led coalition airstrikes, and are poised to enter the main city.
The declaration came after the commander of an Iraqi paramilitary organisation dominated by Tehran-backed units said on Sunday it was willing to send forces into Falluja if efforts to retake the city were too slow.
Abu Mahdi al-Mohandis, military commander of the Hashed al-Shaabi militias, said its forces had surrounded Falluja and left breaching operations to counterterrorism forces, but that could change if the fighting dragged on.