President Joko Widodo agreed to accept worldwide help on Sunday night, Thomas Lembong said.
The wave hit Palu on Friday evening as Muslim worshippers were gathering for evening prayers in local mosques.
The quake and tsunami knocked out power to the island and left many residents without access to clean water and medical supplies.
Indonesian rescue team carry the body of a victim in a major natural disaster and tsunami in Palu, Central Sulawesi, Indonesia, Monday, Oct. 1, 2018.
Exhausted survivors scoured make-shift morgues for loved ones, and authorities struggled to dig out the living or assess the scale of the devastation in more remote regions beyond Palu. According to Sutopo, this is a temporary measure to stop disease spreading.
Dozens of people were reported to be trapped in the rubble of several hotels and a mall in the small city of Palu, 1,500 km (930 miles) northeast of Jakarta, with hundreds more feared buried in landslides that engulfed villages.
In addition to travel and communication problems, relief agencies say their response to the latest crisis is somewhat limited by the fact that many staff members remain deployed to Lombok - an island hundreds of miles to the south of Sulawesi, where a 6.9 magnitude quake struck one month ago.
Indonesia, which is on the seismically active Pacific Ring of Fire, is all too familiar with earthquakes and tsunamis.
Some survivors have been looting shops for food, water and fuel, telling reporters they have run out of supplies.
"The government, the President have come here, but what we really need is food and water".
The disaster agency said a tsunami warning system, which might have saved lives, had not worked for six years due to a lack of money.
Disaster agency BNBP said it had "limited data, information and access" to the affected areas.
One survivor, Adi, was hugging his wife by the beach when the tsunami struck on Friday.
"My sister was found embracing her father", he said. "I was carried about 50 metres".
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Local military spokesman Mohammad Thorir said the area near a public cemetery could hold 1,000 bodies.
"Our funding has been going down every year", Nugroho said.
Indonesia's President Widodo has approved foreign aid after visiting the area on Sunday.
But Nugroho indicated sheer power of will may not be enough.
The Lippo Group said in a statement that it has delivered food, drinking water and clothing worth more than Rp 365 million, collected from Unilever Indonesia, OrangTua Group, Sumber Pangan Sejahtera, Coca-Cola, Mayora, Indomarco, Artaboga and Matahari Department Store, to Palu and Donggala on Sunday.
Almost 50,000 people have been displaced from their homes in Palu alone, Nugroho said, and hospitals were overwhelmed.
One woman was rescued alive from ruins overnight in the Palu neighborhood of Balaroa, where about 1,700 houses were swallowed up by liquefied soil, the national rescue agency said.
The trench can be enlarged if needed, said Willem Rampangilei, chief of Indonesia's National Disaster Mitigation Agency.
At least five foreigners, including three French nationals as well as a Malaysian and the South Korean national, are unaccounted for, Nugroho said. He also acknowledged problems with getting aid to the region, and urged people to be patient.
Satellite imagery provided by regional relief teams showed severe damage at some of the area's major ports, with large ships tossed on land, quays and bridges trashed and shipping containers thrown around.
A collapsed bridge is pictured in Palu, in Indonesia's Central Sulawesi province, on October 1, 2018. Roads were blocked and communications disrupted with other cities.
Indonesia, home to 260 million people, is one of the world's most disaster-prone nations.
Indonesia, one of the world's most disaster-prone countries, lies on the Pacific "Ring of Fire" where tectonic plates collide and numerous world's volcanic eruptions and earthquakes occur.