Yes, a J-1 visa holder is eligible for a green card. To obtain a green card, the alien must first file a petition I-140 with the USCIS. After this, it is essential to apply for an Adjustment of Status (AOS) or for an immigrant visa. If the alien resides in the U.S., they should use an AOS application. If the alien lives in a foreign country, they need to apply for an immigrant visa.