MEA spokesperson Syed Akbaruddin informed reporters that said the first lot of 19 had been taken to Talaimannar in Sri Lanka's northwest, and the Indian consul in Jaffna was trying to gain access to them.
"We are trying to meet the Sri Lankan authorities regarding the other 34 too," he added.
He said Sri Lanka's deputy high commissioner was summoned on Thursday and told to treat the arrested men in a humane manner and that there should be no violence against them.
The spokesman said India expected these fishermen to be swiftly sent back to India along with their fishing vessels.
Earlier, the Sri Lankan Navy said it had arrested 53 Indian fishermen who entered Sri Lankan waters illegally.
Navy spokesman Kosala Warnakulasuriya said 19 were caught off the northern seas of Mannar on Wednesday evening and 34 off Delft islands, also in the north, on Thursday.
Indian and Sri Lankan fishermen often stray into each other's territory as both countries are divided by a small strip of sea.