Not every U.S. citizen wishing to obtain a green card for a foreign-born family member actually lives in the United States. Some, for example, get married while living overseas, and some are members of families that are based abroad. At some point, however, the citizen and immediate family may wish to move to the United States.
This raises the question: Can an overseas U.S. citizen sponsor family members for U.S. green cards (U.S. lawful permanent residence)? It's possible, but it won’t be straightforward.
First, understand that even a U.S. citizen cannot petition for (sponsor) all categories of family members for a U.S. green card. The only relations allowed immediate admission (without spending years on a waiting list) are immediate relatives of U.S. citizens; that is, parents, a spouse, and unmarried minor children (including stepchildren if the parents' marriage took place before the child turned 18).
A citizen can petition for certain other family members (adult children, married children, and siblings), but the wait for their "priority date" to become current will be long. See Green Card Through a U.S. Family Member: Who Qualifies? for details.
To start the process of obtaining green cards for family, the citizen will need to prepare and submit Form I-130 to the appropriate U.S. government agency. (See, for example, Preparing an I-130 Visa Petition for the Immigrating Spouse of U.S. Citizen or Filling Out and Submitting Form I-130 for Parents of a U.S. Citizen or Filling Out, Submitting Form I-130 for Unmarried, Minor Child of U.S. Citizen or Permanent Resident.
After that petition is approved, the family members would need to prepare additional forms and documents, and eventually attend an interview at a U.S. embassy or consulate in their country of residence.
Here is where things get sticky. Every intending immigrant must prove that he or she is not “inadmissible” to the United States. U.S. immigration law contains numerous grounds of inadmissibility, barring noncitizens from entry for security, criminal, health, financial, and other reasons.
Specifically, applicants are expected to prove that they will not likely become a “public charge” (dependent on need-based government assistance). To prove this, the U.S. petitioner must sign paperwork promising to act as financial sponsor in the United States. The paperwork in question is a Form I-864 Affidavit of Support.
But only someone who is “domiciled” in the U.S. (living there) is eligible to serve as a sponsor. A U.S. citizen who doesn't plan to move to the U.S. can (and must, by law) still fill out a Form I-864, but it's not enough to overcome this ground of inadmissibility.
You will most likely need to find someone within the U.S. to serve as a joint sponsor. This is no small thing to ask of someone, as described in What Sponsors Should Know Before Signing Form I-864 Affidavit of Support.
If the U.S. citizen does plan to move to the U.S. with the rest of the family, it will be much easier to overcome. The citizen sponsor will need to provide evidence of these plans, however, ideally including a job offer.
It would be wise to consult an experienced U.S. immigration attorney, preferably in the state where the immigrating family members intend to live or where a likely joint sponsor resides.