Police in Fresno, California have been working for two weeks to try and retrieve the vehicle, which is stuck in the middle of a fast-moving part of Kings River.
In a statement, Fresno County Sheriff's spokesman Tony Botti said search and rescue teams had confirmed "there are two people in the car" but police have not been able to identify them.
"Recovering the bodies is a top priority, but safety for our personnel is also a top concern," he said. "A date of when this will happen is still to be determined."
Thai authorities said they believe the couple in the car to be two Thai students who have been missing since late July.
According to police, park rangers and highway patrol officers found traces of a car accident at the edge of a sheer cliff at the edge of a freeway heading into King's Canyon National Park on July 26.
The car had fallen from the cliff into raging waters made deeper and more violent by summer snow melt. A helicopter flyby of the crash showed the vehicle standing half upright in the water, with all airbags fired, preventing police from seeing inside.
Police have been monitoring water levels to try to find an optimal time to retrieve the car, and have flown a drone over the scene of the accident to get a better view inside, with no luck so far.
Two days after the car was spotted, a local hotel told police that two Thai tourists who had driven out in the morning had never returned, and asked officers to check if they were involved in the car accident.
In a statement, the Thai consulate in Los Angeles said it was working closely with police, and has informed the families of the missing students, who have not been formally identified.
The consulate said members of the Thai community in California have expressed frustration over the slow process of retrieving the car, and have threatened to stage protests in Los Angeles and Fresno on August 14 if a recovery operation has not taken place by then.
A consular official told CNN that some family members have already traveled to California, but they have no idea when the bodies may be recovered.