J1 visa holders and their dependent a ate required to leave the country at the end of their program within 30 days. Must remain out of the country for two years before re-entry. Can apply for H1-b visa at that time. Do consult an experienced immigration attorney for guidance
If your husband's J visa states that he is not subject to the 2-year requirement, then he most likely will not be required to leave the US based on the J visa.
However, your husband must remain in some valid immigration status.
You should file an I-130 Petition for your husband.
If you have been a permanent resident for at least 4 years and 9 months, then you should consider filing for naturalization. If you become a US Citizen, that will help your husband.
I strongly suggest that you consult an experienced immigration attorney.
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What really controls this question is how many years you've been an LPR and when you can apply for citizenship. If you're a U.S. citizen, his filing is considered an immediate relative case and will be processed within 4-5 months.
If you are not close to becoming a US citizen, your husband should consider visa alternatives like F-1, H1B or some other non-immigrant visa. But for most visa types he will need a US employer to petition him.
You don't have to wait to file the immediate relative petition, but the approval of this petition will not allow him to stay in the US. He will still need to consider another visa alternative in order to stay in the US, while visa numbers become available for his immigrant relative case.
Please consider talking to an experienced immigration attorney.