Meanwhile, local distributors in Israel are shunning the film, which Jewish groups say demonises Jews by depicting them as pressuring the Romans into crucifying Jesus.
The film has banked more than $315 million since its release in February.
Only one per cent of Palestinians in the West Bank and Gaza Strip are Christians while the other 99% are Muslims, who revere Jesus as a prophet but do not believe he was crucified.
The portrayal of a prophet in a film is forbidden under Islam.
"People are calling me from everywhere in the West Bank –from Bethlehem, Hebron, Ram Allah and Nablus - to ask for copies of the movie," said the owner of a Gaza city video shop, which sells pirated copies of new release movies.
The Passion of the Christ has outsold other Hollywood blockbusters in Gaza and the West Bank's pirated video market, including Matrix Revolutions and The Last Samurai.
In Israel, the local agent for the film's international distributor Icon Entertainment said it passed on its option to show The Passion of the Christ, but declined to specify its reasons other than to say the movie was "sensitive".
Industry insiders in Israel say local distributors are not interested in the film because of allegations it is anti-Semitic and for expected lukewarm audience response.
Jewish groups and some Roman Catholic clerics have expressed concern the film could foment anti-Jewish attacks.
Palestinian President Yasir Arafat watched a preview of the film at his West Bank headquarters earlier this month. Aides said he found the film "moving".