The fishermen were arrested on charges of crossing the International Maritime Boundary Line (IMBL) when they were fishing near Katchatheevu. One fishing boat was also seized by the Lankan navy.
Chief Minister K Palaniswami said the incident came "close on the heels" of Lankan navy personnel allegedly killing an Indian fisherman on March 6.
"Although shortly after the shooting incident, 85 Indian fishermen were released by the Sri Lankan Navy, the apprehensions made again today indicate that the Sri Lankan Navy has little regard for the diplomatic efforts being undertaken to solve the vexatious issue," he added.
Tamil Nadu has pointed out various examples where countries sharing "narrow waters between them" have worked out mutually acceptable agreements on fishing through bilateral diplomatic efforts, the chief minister said.
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"The continuous arrests and apprehensions by the Sri Lankan Navy, however, seem to be an attempt only to prevent any such diplomatic solution to the issue," he said and called for effective action from New Delhi to curb the "repeated apprehensions and harassment" of the fishermen by Sri Lanka.
"A decisive shift in the manner in which this sensitive issue is being dealt with is long awaited and is the need of the hour," he wrote in the letter to the Prime Minister.
The Chief Minister sought Modi's direction to External Affairs Ministry officials for concrete action through diplomatic channels to secure the immediate release of the 10 fishermen and 129 boats.
He also reiterated Tamil Nadu's demand of a Rs-1,650 crore package for deep-sea fishing from the Centre.