The document has been found in the east of the Israeli capital and appears to threaten "revenge" on Christians.
It is signed by a group calling itself "ISIS [Islamic State] Palestine", according to local press.
"Vengeance" will be brought upon followers of Christ, the leaflet said.
Christian journalist Paul Calvert is in Israel and told Premier there was "a lot of fear concerning IS" amongst the Christian community.
Recent figures suggest there are 12,000 Christian residents in Jerusalem, and many more Christian tourists every year.
It's thought a large number of Palestinians have left the area to go and fight for Islamic State in Iraq and Syria, although there is no way to verify this.
IS has made vocal attacks on the Christian faith in the past, with one its magazines even showing the group's black flag flying above the Vatican.
Aryah King, a Jerusalem City Councillor, said Islamic fundamentalism was "waking up" and becoming "like a trend".
"We've seen more Muslims, not just in Jerusalem, but in Northern Israel, who have gone to join IS," he said.
He warned that IS was attempting to create a presence in Israel.
In December a militant from Islamic State said the group would not stop until Jerusalem had fallen: "Convert to Islam or pay the ultimate price when your armies are expelled from Muhammad's peninsula, from Jerusalem, and all Muslim lands."
The jihadi added: "The Jews and the Crusaders [Christians] are scared and weak".
Elias Kasabre, a man living in Jerusalem, translated the leaflet for Premier.
It says: "Our people Muslims, we salute your resistance and your jihad and we ask you to report on the locations for Christians living among you and who are working clients for the Zionists and the west and their traitorous associations that work to spread debauchery and secularism and keep Muslims away from their religion.
"Islamic state know where they live. Most of these Christians have been spotted but not all of them.
"Strike forces, swords and arms group will work to cleanse the Islamic neighbourhoods of these Christians during the holy month of Ramadan, so when Eid al-fitr comes they will become completely clean.
"We will begin with neighbourhoods of Beit Hanina then Shu'fat then move to the rest of the neighbourhoods in order to get to the old city and Resurrection church to purify it from who betrayed the age era and tell the Christians heathens you must leave immediately or you will get murdered like sheep with the pre-eminence of Eid -el fitr.
"One month more than enough to leave."