Local media reports the downpour has caused flooding that has also damaged homes and bridges.
This has prompted calls for the authorities to open evacuation shelters after residents were ordered to leave the area.
A state emergency management official says the rains have led to the cresting of the Kaupakalua Dam on the island's northern region of Haiku.
Governor David Ige says evacuation shelters have opened at a community centre and a high school.
Officials had warned that if the dam failed that would result in life-threatening flooding as well as significant property damage in areas downstream.
Mr Ige tweeted that Maui County officials have been informed that Kaupakalua Dam has been breached, adding that evacuations had begun.
Maui Mayor Michael Victorino said six homes were damaged or destroyed after the dam flooded the Haiku area.
Two bridges were also destroyed, Fox affiliate KHON2 News said on Twitter.
The dam, dating from 1885, is 17.4 metres high and 122m long, and belongs to the East Maui Irrigation Co.
The owners received a deficiency notice about the dam last year from the state, KHON2 News said in a report. It was one of nine cited as being in poor or unsatisfactory condition.
The state land department is charged with inspecting about 135 registered dams statewide.
In 2006, seven people died after the Ka Loko dam collapsed on the island of Kauai.
Photo NBC News