The new location could be Trivandrum in neighbouring Kerala, the Sunday Times said.
Threats and intimidation of Sri Lankan nationals and Sri Lankan property have been noted in Chennai.
The paper claimed that even the Sri Lankan defence attache based in Chennai had come under harassment.
Last week, two Buddhist monks came under attack triggering noisy protests here near the Indian High Commission premises and Sri Lanka's national carrier also suspended flights to Chennai.
Top Indian officials here met with senior Buddhist monks to try to allay fears.