India's top court has allowed an Italian marine to return home while international arbitration takes place over a 2012 fatal shootings of two Indian fishermen in which he and a fellow marine are implicated.

Salvatore Girone had been free on bail even before Thursday's Supreme Court order but had earlier been asked to remain in New Delhi.

The case has put a strain on relations between India and Italy. Both countries disagree on the facts of the case and over who has jurisdiction.

India has accused Girone and Massimiliano Latorre of killing the two fishermen in Indian waters while they were assigned to anti-piracy duties aboard an Italian commercial ship.

Latorre has been in Italy since 2014 for medical treatment after suffering a stroke in India.