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In theory, yes. But, many people will not be able to do this as a practical matter – or at least, not without a waiver requiring considerable additional effort and expense.
Being in the U.S. without status does not automatically prevent a foreign national from processing for a green card through an embassy or consulate abroad. However, if a foreign national has accrued sufficient “unlawful presence” in the U.S., the act of leaving to go to an overseas U.S. embassy or consulate may subject the person to a bar to reentry (see our FAQs on the three- and ten-year bars).
The foreign national would either need to wait out the period of the bar while overseas before obtaining permanent residence and reentering the U.S., or obtain a waiver of the bar based on a high degree of hardship to a U.S. citizen spouse or parent (but NOT to a child).
There is a way to obtain a decision on the waiver before having to leave in many circumstances, so that an individual knows before departing that the three or ten year bars won’t be an issue (of course, this doesn’t preclude the consulate form raising some other ground of inadmissibility – so departing to process abroad can still be a daunting proposition).