I have a couple right now that is looking to get married. The man is not a legal resident - he is an undocumented immigrant to the United States. He has been here since 2001. As far as he knows, he can't get married here in the USA, so they were thinking about going back to Mexico, getting married, and then having her petition for his citizenship. She was born here in the USA. The question I have is, can they do that? Would INS approve something for someone who's been living here for so many years without documentation?

Not enough information.

It depends on his manner of entry and what he has filed or others have fiiled for him prior to April 30, 2001 to be eligible for AOS under section 245(I). Any criminal arrests, charges or convictions is also important.

Did he enter with a visa and overstay? EWI and file a LC? PC passport? False claim to citizenship?

Get this information and consult with an experienced immigration attorney.

He entered illegally through the border and has not filed anything. He has no criminal arrests or charges of any type.

It will depend on the specific facts of the illegal entry.

If he entered EWI, he will need to use 245(I) to adjust status in the US. 245(I) allows for adjustment with payment of a $1000 penalty to forgive the EWI and unauthorized employment in the US. He will have to have filed a LC or someone filed a preference petition prior to 04/30/2001, the day the extension of 245(I) ended.

If he entered under false pretenses, he may be eligible for a waiver depending on the hardship to his USC spouse.

Strongly recommend consulting with experienced immigration attorney to discuss in detail.