These orders were set forth through a letter after prayers on Friday. The rebel group, which is Sunni, issued the harsh orders in a document obtained by a local media agency.
The document says that an order had been issued after the leaders of the Christian community did not attend a meeting that ISIS had called.
The group, in response, wrote in the letter that all Christians are required to either convert, pay a tax known as jiziya for non-Muslims or as a last resort face death.
Mosul is the second biggest city in Iraq and was overrun by ISIS and rebel groups in June.
The army units from Iraq stationed there, most of whom are Shia, fled when the group moved from Syria over the border and attacked northern Iraq.
Prior to the attack, the Christian community in Mosul was estimated to be 3,000. Many already are believed to have fled Mosul as part of a mass exodus from the city of 33% of the overall population.
Shops that were Christian-owned and Churches in Mosul have been reportedly vandalized by the people who fled.
Abu Bakr al Baghdadi, the leader of the Islamic State held a sermon two weeks ago inside the grand mosque in Mosul. He called on all Muslims to unite in support of his organization.
ISIS is now called the Islamic State, and issued demands that were similar in areas it controls inside Syria. The group posted pictures where Christians were crucified for not following orders in the city of Raqqa.
Iraq’s church leaders have not made any official response to the threats.
The UN assistance mission head in Iraq, Nickolay Mladenov condemned the Islamic State order.
Mladenov said any persecution of any minorities group constitutes a crime against humanity and urged all sides to make sure civilians were protected.