The exact price of the purchase is unknown, but it is thought that Ellison will pay between $500m (£318m) and $600m for 98% of Hawaii's sixth-largest island.
Lana'i, a 141 sq mile (365 sq km) island to the south of Honolulu, is being sold by billionaire David Murdock.
Commonly known as the "Pineapple island", Lana'i is home to around 3,200 residents, along with several luxury holiday resorts.
It was acquired by Murdock in 1985 when he took over the near-bankrupt Hawaiian firm Castle & Cooke, before turning around its assets to make himself hugely wealthy.
In a press statement on Wednesday (June 20), Hawaii Governor Neil Abercrombie confirmed the bid by Ellison.
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"It is my understanding that Mr Ellison has had a long-standing interest in Lana'i," he said.
"He is also a businessman whose record of community involvement in medical research and education causes is equally notable. We look forward to welcoming Mr Ellison in the near future."
After co-founding Oracle in the late 1970s, Ellison helped guide the firm to become one of the world's leading enterprise software companies.
He has an estimated net worth of $36.5 billion, making him sixth on the Forbes list of global billionaires.
The remaining 2% of Lana'i is not owned by Murdock's Castle & Cooke, which remains the primary employer on the island, owning hotels, golf courses and various utilities.
Prior to news of Ellison's bid interest, the Mayor of Lana'i Alan Arakawa expressed his hope that the island's new owner would show as much care and compassion as Murdock.
Speaking to Maui News, he said: "I'm hoping that whoever buys the island will have as much compassion for the residents there as David Murdock. I'm hoping they will be conscientious owners."