There is not a method by which a church can sponsor someone for citizenship. In order to become eligible for citizenship, a worker at a church first would need to become a Lawful Permanent Resident (get a "Green Card"), and later become eligible for naturalization to become a citizen.
There are several types of visas that might be available for religious workers depending upon the circumstances. For example, "R" nonimmigrant visas are for certain ministers or others working in a professional capacity in a religious vocation or occupation, and Fourth Preference "Special Immigrants" is an immigration classification for certain religious workers.
Determining eligibility and establishing an immigration legal strategy can be complex both for a church and for its employee, and great care must be taken in order to be successful.
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David N. Soloway
Frazier, Soloway & Poorak, PC
1800 Century Place, Suite 100
Atlanta, Georgia 30345
404/320-7000 877/232-5352 firstname.lastname@example.org
There are certain nonimmigrant (temporary) visas specifically for religious workers and clergy, although these are very strict categories and not everyone will fit into them.
In addition, to get in to this type of status, the person would need to be abroad, be here in valid legal status so as to be able to change status, or if they are here without status, be able to leave to get the new visa abroad and return without incurring a bar to coming back by leaving after building up enough unlawful presence here.
The short answer is, your church and this individual need to discuss this with an immigration lawyer to see if the person is would qualify for an appropriate visa - and if so, whether there is anything else preventing them from obtaining that visa.
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