The victim, whose name is not being released, was trying to photograph the animal before the elephant attacked him, officers told the BBC.
The tourist, who was 67 years old, and two guides had illegally entered a closed area of the jungle in the state of Tamil Nadu, the British broadcaster said.
The elephant charged the three men and hit the British tourist on the head with its trunk. The two guides escaped, police said.
The victim was taken to a hospital where he was later pronounced dead.
Police said they will arrest the two guides.