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Confirming it had received Marsh's letter, Sri Lanka Cricket president Upali Dharmadasa stated, "We terminated his services and paid him whatever that was due to him.
"It is now a legal matter and we have referred it to our legal department to check it and deal with it."
Marsh was appointed coach in September last year and his contract was terminated after the tour to South Africa where Sri Lanka recorded their maiden Test win. But they lost the three-Test series 2-1.
Marsh's sacking was criticised, with former Sri Lanka captain Arjuna Ranatunga calling the move "disgraceful" and "unprofessional".
Marsh was replaced by South African Graham Ford, who became Sri Lanka's fourth coach since Trevor Bayliss quit after the World Cup final in April 2011.