Costandinos Tselios, aged about 35, who had come to Kodaikanal in 2007 as a tourist, had fallen in love with a local woman and had married her.
They had settled in Adukkam and were doing farming. The couple had a two-year-old girl.
It was not known whether Costandinos had obtained visa to stay in India, police said.
They also said Costandinos had run into financial problem and had sought some loan from his relatives in Greece to take care of his farming work. He was allegedly frustrated as he did not get any reply from them.