The three "did not understand the relic's value" and "cannot remember where they threw away the precious loot", the Ansa news agency said, citing the police.
Police recovered the object's metal frame but could not find the cloth, believed to be part of the robe the pontiff was wearing when he was shot in an assassination attempt in St Peter's Square in 1981.
It was stolen on Saturday along with a cross, which has also been found, from the San Pietro della Ienca church in the mountainous Abruzzo region in central Italy where the late pope loved to go on skiing holidays.
The PrimaDaNoi.it Abruzzo news website said two of those arrested are 23 and 24 years old and are drug addicts known to the police for other petty crimes.
The relic is a framed, tiny square of cloth.
The relic was given to the small church in 2011 by Stanislaw Dziwisz, a cardinal who served as John Paul II's personal secretary until his death in 2005.
Local media said some 50 police officers and sniffer dogs were deployed in the search for the missing relic in an area famed for its weathered stone houses and the little church where the head of the Roman Catholic Church once reportedly took refuge in a storm.
John Paul II and the Italian pope John XXIII, known as "Good Pope John", are set to become saints in a ceremony at the Vatican in April — an event which will substantially increase the value of the stolen relic.