Jayalalithaa presented a detailed memorandum to Modi during their 50-minute luncheon meeting at her Poes Garden residence here.
She called for retrieval of Katchatheevu islet from Sri Lanka, besides stressing on India's role to ensure welfare of Lankan Tamils.
The Chief Minister sought central intervention in matters like the Cauvery and Mullaperiyar disputes with Karnataka and Kerala respectively.
In the memorandum, which was released by the state government, Jayalalithaa demanded that the Centre should immediately form the Cauvery Management Board and Cauvery Water Regulation Committee for implementing the final award given by the disputes tribunal.
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She also raised the Mullaperiyar issue in which her government had succeeded in a court case for raising the storage level to 142 feet.
Jayalalithaa said there could be a "security threat" to the dam structure and its appurtenant structures "from unruly mobs against whom the Kerala Police, instead of taking action, are acting as mute spectators".
"It is, therefore, imperative to deploy the Central Industrial Security Force to guard the Mullaperiyar Dam and its appurtenant structures," she said.
She recalled having suggested a "radical approach" where levy, collection and appropriation of the substitutes for VAT, Central Excise Duty and Service Tax within a state could be delegated completely to the state machinery, with the central machinery focusing on inter-state taxation.
She said a manufacturing state like Tamil Nadu will "permanently lose substantial revenue" if GST is implemented, due to the shift of levy from the point of origin to the point of destination.