Police have been called in to investigate the death last week of seven-year-old Komala at Mysore, in southern Karnataka state, zoo director Manoj Kumar told AFP on Tuesday.
Two elephants and a lion-tailed macaque, an endangered species, were poisoned in August.
"There is a suspicion that some disgruntled employees may be behind the deaths," he said, adding it was still too soon to suspend anyone at the zoo.
A government-run veterinary institute found that zinc phosphide, a cheap pest control agent normally used again rodents, had killed the first two tuskers.
"There is a pattern in all the three deaths. It is due to poisoning," Kumar said.
Komala had been picked out to be sent to Armenia as a gift from Indian President A.P.J. Abdul Kalam.
"Komala was due to be flown out to Armenia on October 14 and there was a delay from the Armenian side. Later the date was revised and fixed for October 30. It is unfortunate," the director said.
The century-old Mysore zoo houses 1,100 animals.