The tribunal's decision, approving an Italian request, was expected to be officially announced in The Hague on Tuesday.
India has accused two marines, Salvatore Girone and Massimiliano Latorre, of the shootings in 2012 while they were assigned to anti-piracy duties aboard an Italian commercial ship, the Enrica Lexie.
Latorre has been in Italy since 2014 for medical treatment following a stroke suffered while detained in India. The Italian government had insisted that Girone also be allowed to return to Italy, for the length of the arbitration process, which could take years.
Italy contends the incident occurred in international waters. It disputes India's claim to jurisdiction, insisting the case should be handled by the Italians. The Hague tribunal has been asked to rule on the jurisdiction dispute.
India has been bullish in insisting its sovereignty in the case be recognized. Its government didn't immediately issue any official comment on Monday's development, or indicate when Girone might return.
"The conditions of (Girone's) return will be agreed upon by Italy and India," Italy's foreign ministry said in a statement.
In the evening, Italian Foreign Minister Paolo Gentiloni expressed satisfaction.
Girone "won't be back tomorrow, it will take some weeks" for him to set foot in Italy, Gentiloni said in an appearance on state TV. "But an important decision was taken."
"Today's decision makes me think we did well to choose an international judge," Gentiloni added. He said that when the
Earlier, the Italian foreign ministry said in its statement: "The government is counting on a constructive attitude by India."
Earlier in the dispute, the two marines were allowed to return home to see their families on holiday, but then were sent back to India under the terms of that arrangement.
(AP)