Foreign Ministry spokesman Marty Natalegawa told CNN it appeared the blasts were "the work of terrorists."
Australia's Foreign Minister Alexander Downer told Austrialia's ABC radio that one Australian was among those killed.
The blasts happened in Jimbaran and Kuta, the top tourist spots on the island, and Natalegawa said reports indicated the blasts appeared to have occurred just minutes apart. (Watch video: Bali rocked by blasts)
Video of the scene in Kuta showed the glass windows of several upscale stores shattered, with glass littering the street, and the awning of Raja's Bar and Restaurant blown askew.
There have been local media reports that Westerners were among the casualties.
Kuta is where two powerful bombs detonated near several nightclubs on October 12, 2002, killing 202 people, most of them Australian and Indonesian.
That attack was blamed on Jemaah Islamiyah, which has ties to Osama bin Laden's al Qaeda terrorist network.
Indonesian President Susilo Bambang Yudhoyono recently issued a warning that terrorist cells inside the country were still active, despite hundreds of arrests.
Shortly after Saturday's blast, Yudhoyono addressed the nation on state television, expressing his concern and condemning what he called a terrorist attack.
He said he had received intelligence information in July about terrorist elements in Malaysia and Philippines planning to target Indonesia, and had increased security in Jakarta and Bali as a precaution.
The blasts also come after recent warnings from several other governments, including the United States, of a high terrorist threat to foreigners in Indonesia as the holy Muslim month of Ramadan arrives early next week.
A report issued in early September warned that bin laden and his top lieutenant Ayman al-Zawahiri may be planning a series of attacks in October, dubbed "The Great Ramadan Offensive."
On Friday, the U.S. Embassy in Jakarta reiterated its warden's message issued in May, which said the threat of terrorism is high and that Americans there should be vigilant.
"Attacks could occur at anytime and could be directed against any location, including those frequented by foreigners and identifiably American or other western facilities or businesses in Indonesia," the message said.
The island of Bali -- a popular spot for tourists across the world -- is actually a Hindu enclave in Indonesia, which is the world's most populous Muslim nation.
Natalegawa said he was confident authorities would find the attackers.
"We know what it takes to bring the perpetrators to justice because our success rate ... has been rather good," he said.
