Malaysia's most wanted terrorist, Dr Mahmud Ahmad, is believed to have been killed in fighting in Marawi on Mindanao island.
It was reported that Philippine troops, with the help of civilians, are trying to recover Mahmud's remains to validate the intelligence received.
Galvez said there are "strong indications" that two or three of the Maute brothers - among the key players in the siege - had been killed, including Omarkhayam Maute, believed to be the group's top leader.
Gen. Eduardo Ano told The Associated Press that Malaysian Mahmud bin Ahmad reportedly channeled more than $600,000 from the IS group to acquire firearms, food and other supplies for the attack in Marawi.
Further raising concern in the Philippines and regionally, another militant group, the Bangsamoro Islamic Freedom Fighters (Biff), staged a surprise raid earlier this week on a village 190km south of Marawi, underscoring the challenge of reining in the spread of Islamic militants in the southern Philippines.
Isnilon Hapilon, a veteran Filipino militant said to be the leader of the Islamic State (IS) group in Southeast Asia, has not been seen in the battle zone in Marawi City, said Lt. Gen. Carlito Galvez, head of the military's Western Mindanao Command.
Eid al-Fitr celebration to kick off in Taipei on Sunday
While in Bangladesh, India and Pakistan Ramadan did not begin until May 28 and therefore will for the Eid moon on Sunday night. Because Eid takes place at the same point in each lunar year, the date of the festivity varies on the Gregorian calendar.
The militants in Indonesia and Philippines were encouraged by series of terror attacks in Bangladesh from early 2013 to last year's July 1 Gulshan cafe attack and they wanted to follow the model of the attacks in their home countries, according to a recent report of Indonesia-based Institute for Policy Analysis of Conflict (IPAC). "Be that as it may, if there are any complaints, our armed forces are open to receiving these complaints and acting on them immediately", said the presidential spokesperson. He said three boatloads of gunmen who tried to join the militants were blasted by navy gunboats three days ago in Lake Lanao, which borders Marawi.
Ano said that the battle was taking longer because the militants were using civilians as human shields.
The Philippine English daily quoting Kyodo News, a Japan-based news agency, said the information was revealed by Philippine officials during a meeting with Indonesian and Malaysian counterparts in Manila last Thursday. But if confirmed, his death would be a blow to efforts by the Islamic State in Iraq and Syria (ISIS) group to gain a firm foothold in the war-torn Philippine island of Mindanao.
Defense Minister Marise Payne said Australia will work with its partners in Southeast Asia to combat terrorism, as it is also a direct threat to her country.
Philippine President Rodrigo Duterte declared martial law in Marawi and the entire southern region of Mindanao, unleashing an offensive to crush what he said was an attempt by the jihadist group to establish a province in the area. "It's much broader problem that needs a strategic solution".