Victims accused Barros of having witnessed the abuse but doing nothing to stop it. Barros has denied the allegations.
Barros has been at the center of the scandal ever since Francis appointed him bishop of Osorno in 2015 over the objections of the local faithful, his own sex-abuse prevention advisers and some of Chile's other bishops.
In a letter released on April 11, Pope Francis said he had been mistaken in his assessment of the situation in Chile, and he begged for forgiveness from survivors and others he offended.
Barros, who was a protege of Karadima, is among those accused of witnessing and ignoring his wrongdoing.
Pope Francis remains in the spotlight, following recent comments about climate change with pleads to oil executives that call for clean fuel.
Jaime Coiro, general secretary of the Chilean Catholic Church, noted that the pope has said the case will require short-, medium- and long-term measures, which could include accepting the resignations of more bishops.
"Today begins a new day for the Catholic Church in Chile and hopefully the world", Cruz wrote. He has been sent back to Chile to gather more information.
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Barros, who is a central figure in the controversy, has been accused of covering up abuse committed by another priest in the 80s and 90s.
Barros, Bishop Gonzalo Duarte of Valparaiso and Bishop Cristian Caro of Puerto Montt tendered their resignations last month along with Chile's remaining 31 bishops. However, the pope, who hails from Argentina, initially defended the prelate in January, saying the allegations were "slander".
He also chose to host three Chilean sex abuse survivors at his home in the Vatican so he could apologise to them personally and hear their recommendations for change.
Those findings, which leaked to the media while the Chilean bishops were at the Vatican, have opened a Pandora's Box of new accusations that recently led Francis to become the first pope to refer to a "culture of abuse and cover-up" in the Catholic Church.
In 2011, Fr Karadima was sentenced to a life of prayer and penance by the Vatican after he was found guilty of sexually abusing boys.
Francis has apologised to the victims and admitted he had made "grave mistakes" after reading a 2,300-page report on abuses in Chile.
Juan Carlos Claret, spokesman for a group of lay Catholics in Osorno who fought for Barros' removal from office, said it was a "minimum condition" that they had sought from the pope.
Since 2000, about 80 Catholic priests have been reported to authorities in Chile for alleged sexual abuse.