A member of the public discovered the animals in the trap in the River Tyne, next to Haddington Golf Course.
It is thought the pups swam into the net and the mother followed in a bid to free them.
It is believed that the net was being used illegally to catch eels or crayfish.
The Scottish SPCA is helping police with the investigation.
Gavin Ross, of Police Scotland, said: "Examination of the otters confirms that they were an adult female and two young and it appears they became trapped within the net and were unable to escape.
"Using a net trap for catching eels and crayfish is illegal and we would urge members of the public who remembers seeing anyone fishing in this manner to contact police immediately.
"The death of these animals will be of great distress to many members of the local community, particularly as their return to our rivers is an environmental success story.
"Anyone who has information relevant to our ongoing investigation should contact police immediately."
A police investigation has been launched after an otter and her two pups were found dead inside an illegal trapping net in an East Lothian river.