One person died on Sumatra island and the other in an army hospital in the capital Jakarta, said Runizar Ruesin, the head of the ministry's bird flu information centre.
"They tested positive in the first test," he said, adding a second was being conducted to confirm initial results.
Positive results from two Indonesian laboratories are required for the World Health Organization to include the case in its records.
The deaths, if confirmed, would take Indonesia's death toll from the virus to 71, the world's highest.