The meeting between Modi and Jayalalithaa lasted around 50 minutes.
The chief minister also hosted lunch for Modi.
She said the Tamils' distinct identity and cultural presence in Sri Lanka, particularly in the northeast, was the guiding principle behind the India-Sri Lanka accord of 1987.
That accord, which led to the deployment of Indian troops in Sri Lanka, led to the 13th amendment to the Sri Lankan constitution.
"The process of securing the right to self determination, through democratic decentralization by the 13th amendment ... should be the springboard for Sri Lankan Tamils to eventually realise the aspiration of Tamil Eelam," the memorandum noted.
"I strongly urge the (Indian) government to take all possible steps to ensure that the process of democratic decentralization, which is integral to the survival of the Tamils in Sri Lanka, is expedited," she said.
According to Jayalalithaa, this should eventually lead to the Tamils of Sri Lanka realizing their "legitimate aspirations".
"I hope the (Indian) government, as a leader in the region and as a champion of human rights and democracy, will decisively take a bold stand in support of the much discriminated against and long suffering Tamil minorities in Sri Lanka."
She also urged the Indian government to take necessary steps to bring to book those who killed innocent Tamils in Sri Lanka -- by approaching international forums like the UN Human Rights Council.