The daily said a letter from the pontiff was handed to Khamenei hours before Iranian President Mahmoud Ahmadinejad announced the group’s release on Wednesday.
The Vatican confirmed that a letter was sent but did not give details.
According to the newspaper, the pope said he was confident there could be a solution and asked Khamenei to do what he could to ensure that the eight Royal Navy sailors and seven Royal Marines could return home in time for Easter.
He is said to have written that such action would be a “significant religious gesture of goodwill” from the Iranian people.
Ahmadinejad said the sailors and marines were “forgiven” for trespassing into Iranian territorial waters and said the decision was taken “on the occasion of the birthday of the great prophet (Mohammed)… and for the occasion of the passing of Christ”.
The 14 men and one woman were seized near the Shatt al-Arab waterway dividing Iraq and Iran on March 23, leading to a tense diplomatic stand-off between London and Iran. Britain denies they encroached into Iranian territory.
The group returned to Britain Thursday and on Friday disputed Iranian claims that they had been well-treated, telling of psychological and emotional torture during their captivity.