Photographs and videos widely shared online show the young apes sitting together on the forest floor and eating bananas, thousands of kilometres from the tropical rainforests where they belong.
Orangutans are not native to India. The critically endangered apes are found in the wild only in Indonesia and Malaysia.
Odisha's forest minister has said investigators are examining whether the animals were trafficked into the country by an international smuggling network and abandoned near a highway.
The orangutans were rescued early on Tuesday in Balasore district after local people spotted them and alerted forest officials.
They were given first aid before being taken to Nandankanan Zoological Park in the state capital, Bhubaneswar, where they are in quarantine and being monitored.
Balasore District Forest Officer Pratik Prakash Indalkar told the BBC that the animals could not have reached Odisha naturally and may have been transported there by road.
"The five animals were found in Ranakata, a proposed reserve forest in Bhogarai block, near the West Bengal border," he said.
"The area is covered mainly by casuarina trees and is not a dense forest. It is also close to a state highway. "
The Odisha government has also reportedly asked India's Wildlife Crime Control Bureau to investigate. Police, forest officials and a special task force are involved in the inquiry.
