The London Telegraph quotes Monsignor Marco Fibbi, a spokesman for the Vicariate of Rome, as saying that the demand was initially slow, "but last week, [they] received around 5000 requests from worshippers."
The vicariate is sending a tiny shred of one of John Paul's white papal robes to anyone who visits its website, or sends a fax or letter.
The piece of material worn by John Paul is not technically considered a relic, because he has not yet been canonised.
Instead, it is called an "ex-indumentis" or "from the clothing", a phrase that usually refers to a second-class relic. A first-class relic is usually a piece of a saint's skeleton.
Worshippers who receive the relic also get a prayer card, which thanks God for "having given Pope John Paul II to the church and for having made the tenderness of your paternity, the glory of Christ's cross and the splendour of the Holy Spirit shine within him".
Monsignor Fibbi said the church had decided to distribute the items after seeing hawkers in St Peter's Square selling fake relics to tourists.
Monsignor Fibbi said relics would not be sent out to "collectors" or to "people who simply admired the Holy Father".
"We only really want to send out relics to people who intend to use them for prayer," he said.
Rush for shred of Vatican history (The Age, 26/9/07)
Pope's robe cut up for 100,000 'holy relics' (Times Online, 25/9/07)
LINKS (not necessarily endorsed by Church Resources)
Beatification of John Paul II (Vatican Vicariate)
JPII relics now available (CathNews, 17/9/07)