Opponents of the agreement say their concerns include separating elephants from their family, keeping the animals in captivity, and using a creature deemed sacred for commercial purposes.
Safe and other international animal welfare groups such as Born Free and In Defence of Animals have also criticised the gift of Nandi and want the New Zealand Government and Auckland Zoo to turn it down.
Auckland Zoo has previously said Nandi was from an orphanage and would never be released into the wild. Both Sri Lankan and Auckland Zoo experts believed 5-year-old Nandi was the right fit and mature enough to be moved.
Aruna Abeygoonesekera, the honorary consul for Sri Lanka in New Zealand, said as far as he was aware it was the first time the issue had been taken to court. "I think it is probably to be anticipated, firstly because it is the second elephant that is coming out of Sri Lanka to New Zealand that some of these groups have expressed their concerns and gone to court over this, which they didn't before."
Abeygoonesekera helped broker the gift of Anjalee and said it was not uncommon for Sri Lanka to gift an elephant. "It's a very gracious gift and done at a high level to recognise strong bonds between two countries."
Auckland Zoo had "wonderful facilities" to house elephants, he said.
"They are far better looked after than they are in Sri Lanka, I can tell that to you very definitely."
Auckland Zoo director Jonathan Wilcken said the petition was a matter for the Sri Lankan Government.
Wilcken said Anjalee and Nandi were both from a Sri Lankan orphanage so could never be released into the wild. He said Auckland Zoo offered a world class elephant programme - Anjalee had put on 700kg in the past year, was in "exceptional health" physically and mentally and had bonded with Burma, the zoo's 32-year-old Asian elephant.
"Just as we are providing for Anjalee, we can offer Nandi an exceptional quality of life in a safe, secure and happy environment."
Auckland Zoo was given $3 million in Auckland Council funding for two more Asian elephants in 2011. Wilcken said concerns the elephants were used for commercial purposes were misplaced.
"Auckland Zoo does not make money from having elephants but it is through our elephant programme that we can continue and further our support to elephant conservation."
The Sri Lankan Mirror newspaper reported the petitioners also took aim at the zoo's elderly elephant Kashin, who was euthanised after developing foot abscesses and arthritis "common to elephants kept in captivity".
"The petitioners said this would be the fate to which any other elephant brought there would be subjected," the Mirror reported.
Wilcken said Kashin had arrived at Auckland Zoo from an American zoo more than 40 years ago with pre-existing health conditions that Auckland had helped her with throughout her life. "Kashin lived a long and happy life at Auckland Zoo." Kashin was born in captivity in Thailand in 1968 and died aged 40 in 2009.