Yes, there is no prohibition in marrying a person who is out of status.
Your boyfriend is not illegal. First, he entered with inspection and a visa of some sort. He may be out of status and what is referred to as an overstay, but that does not make him illegal. Second, to be considered illegal under immigration law, the attorney general of the US must bring action against him.
Finally, if your intention is marry this man for love, there is no law against it. This happens thousands of times a year. However, if you are marrying him just for the US government to issue him a green card, that is another story. Marriage fraud is against the law and it has severe penalties for both of you.
There is no prohibition on marriage based on status. In fact, your marriage and his use of his passport to enter the U.S. likely means he can get permanent resident status (or a green card as it is commonly called). I suggest you retain counsel as soon as possible as if you are married and you do file papers for him, he'll be in a safer position if he encounters law enforcement. Just don't marry him so he can get a benefit but rather because you want to spend your life with him.
As long as he entered the US on a visa, he can adjust his status through marriage. You should contact an experienced immigration attorney who can guide you through the entire process. Good Luck.
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There is no prohibition against marriage to a person who is out of status. Just make sure that both of you do not have an impediment on re-marriage. If you are a US citizen and since he entered the country with a lawfully issued visa and there are no other issues of inadmissibility, you can file for him to get his green card if you guys do get married. retain an experienced attorney to guide you through the process.
This is not legal advice and a client attorney relationship is not created. For a free consultation call (718)234-5588.