The two marines, in full dress uniform, made no comment to reporters when they arrived at the military prosecutor's office, which had summoned them for its own probe of the 2012 shooting.
India and Italy are in a bitter standoff over the case.
The marines have been charged in India with killing two fishermen. Earlier this week, Supreme Court indefinitely extended its order barring the Italian ambassador from leaving the country after Italy refused to return the marines to India.
Italy's foreign ministry has called that court decision a clear violation of diplomatic relations.
The Italians were on anti-pirate duty aboard a cargo ship off India's coast in February last year when the fishermen, aboard an Indian fishing boat, were killed.
Italy insists the shooting happened in international waters during an international anti-piracy mission and thus, Rome, not India, should have jurisdiction. But a local Indian court insisted it would conduct the probe and the marines were ordered held in India.
The marines did go home to Italy for Christmas holidays and for the national election in February but after that Italy announced they would not return to India.
The Chief Justice of India has expressed outrage, claiming that the ambassador had given his word.
An Italian government official, briefing reporters in Rome on customary condition of anonymity, said Italy had submitted affidavits saying the marines would return to India if requirements in the Italian constitution were respected. The constitution says Italians cannot be tried by a court that hasn't already been established.
The Indian court has ordered the Italians to be tried by a special tribunal but that has yet to be created.
Italy says it has also asked Indian judicial authorities to hand over the official transcripts of the marines' questioning in India as well as the results of ballistic tests, but so far hasn't received them.
Military Prosecutor Marco De Paolis must decide whether the marines should be charged in Italy or whether the case against them should be shelved.