Ministers and religious workers are eligible for an R-1 visa, which would allow them to stay and work in the United States for up to five years. It is also possible to obtain permanent residence through a religious occupation, although the standards are slightly different.
To qualify for an R-1 visa, you will need to show that you have been a member of the same denomination as the sponsoring organization for at least two years, and that you will be coming here to work at least 20 hours a week in a religious capacity. USCIS will also require the sponsoring organization to show that they are a tax-exempt religious organization, provide a list of other workers at the same site (including volunteers), and the size of the denomination. USCIS often oversteps their authority and requests a membership list- this is unconstitutional and should not be provided. Consult with an immigration attorney to be sure that all required documentation is provided, and also remind USCIS of the limits to what they request.