A nine-year-old child and the boat’s captain are believed to be among those who died.
At least four other people were injured in the crash, the coastguard said.
A total of 21 people, including the injured, were rescued from the waters between Aegina and the small deserted islet of Moni.
The tourist boat was transporting holidaymakers going for a swim, according to the coastguard.
A search and rescue operation involving patrol boats, a helicopter and nearby private vessels was continuing to locate more possible survivors.
More than 20 people were believed to have been on board the tourist boat when it sank.
However, the authorities said the precise number and details of how the incident occurred remained unclear as rescue efforts were focused on locating survivors.
It was also unclear how many people had been on board the speedboat, or the nationalities of the survivors.
The four bodies recovered from the water were those of three men, one of them believed to have been the captain, and that of a nine-year-old child.
Aegina’s proximity to the Greek capital, one hour away by ferry, makes it a popular destination for both Greeks and foreign tourists.
Small tourist boats ferry holidaymakers from the island’s port village of Perdika to Moni, an islet inhabited by deer and peacocks which features a popular sandy cove.
Other attractions include the Temple of Aphaea, near the resort of Agia Marina.
The classical ruin has been described as being almost as elegant as the Parthenon in Athens in an olive grove setting.