Authorities have ordered the evacuation of villagers living within a high danger zone that in places extends 12 kilometers from Mount Agung's crater.
But people further away are also leaving, said National Disaster Mitigation Agency spokesman Sutopo Purwo Nugroho.
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Evacuees are taking shelter at more than 370 sites across the island that include temporary camps, sport centers, village halls and the houses of friends and relatives.
Villager Wayan Merta said he was among the first to evacuate last week because his village is just 6 kilometers (4 miles) from the summit.
"We have already sold our cattle, because we thought it was better than leaving them there for nothing," he said.
"My feeling is the mountain will erupt," he said. "But no one knows, we just pray."
Sutopo said it was "natural" that people outside the immediate danger zone are leaving. More than 500,000 people evacuated when Mount Merapi in central Java erupted in 2010, more than double the population in the exclusion zone around that volcano, he said.
In 1963, Agung hurled ash as high as 20 kilometers and remained active about a year. Lava traveled 7.5 kilometers and ash reached Jakarta, about 1,000 kilometers away.
The mountain, 72 kilometers to the northeast of the tourist hotspot of Kuta, is among more than 120 active volcanoes in Indonesia.
The country of thousands of islands is prone to seismic upheaval due to its location on the Pacific "Ring of Fire," an arc of volcanoes and fault lines encircling the Pacific Basin.
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