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H1-b签证已经六年无法续签了，公司还可以帮办绿卡吗？

-Yes. 

若办NIW是否就和签证过期无关了？

-Correct.

You need to have a legal status to stay in US such as F1. Otherwise, you have to leave US and then, apply for the immigrant visa to enter US when your priority date is current in the future either through the company sponsor sponsor or your NIW.

Another choice is that you leave US and stay outside US for one year and then, apply for H-1 visa to return to US if the US company sponsor your H-1, but you have to be selected randomly by USCIS in the H-1 visa lottery unless you work for a college/university or non-profit institution.