I am posting for some friends. My friend has a Green Card in the US and his fiancee has Canadian citizenship and is currently in the US on a 6 month visa, which expires on Sept 30. She is 8 months pregnant so cannot travel on an airplane. She is due to give birth in mid-late October. She has access to free medical care for her and the baby, so is covered from that standpoint. Their 1st step I assume is to do an I-539 to get her application for an extension of stay so that she is not guilty of an overstay.

They plan to marry, but we can't figure out what is the best sequence of events to best ensure that she eventually is allowed permanently into the US. As he has only a Green Card, it seems that he cannot follow the K-1 Fiancee process anyway, as he has not yet started the process toward full citizenship. Should they marry while she is here on her tourist visa, but before the Sept 30 date? Is that Sept 30 date meaningful with respect to the USCIS perception of the status of this marriage? They actually have plans to marry this week, Sept 15, and I'm not sure that is a good or bad idea wrt its impact on their plans for her to come permanently to the US eventually.

K-1 and K-3 are available for USC's only.....

Sorry, I know it's late but maybe I still have time, I just noticed the post:
There are two options:
1. she can overstay the visa, get married soon and apply for AOS. supposedly the overstaying should not matter since she has entered the country legally, that is on a visa. i am a permanent resident but my husband, a USC, says he read comments by immigration lawyers who said that is the case and an overstay should not impact their application as long as you enter the US legally.
2.get married before the 30th and file for AOS, preferably also before the 30th.

I personally would choose the second option.
Good luck and let me know if I can help in any way.

I think he has to wait until he is a citizen to file for her. She should not overstay cause it looks bad in the visa process. She should go home and since he's a permanent resident he can go with her for the birth and return to the US. After he complete the naturalization process he can file for her.

They can get married, and he, as an LPR can file an I-130 for her. The wait time will be a few years waiting for a visa number to become available based upon their priority.

Un fortunately, he cannot wait it out here and should leave before her authorized stay expires.

AOS is not an option.

Thanks all for the advice. We did an I-539 application for her, to extend her stay, as she is going to have her baby in the next couple of weeks and can't fly. She won't hear about the extension for a while, so is planning to take the new baby back to Canada a couple of months after the birth, which will be less than 120 days overstay of her current B-2 visa. If she will be approved for the extension, everything is OK. If rejected, she will have overstayed, retroactive to Sept. 30. This is a risk she has to take due to her pregnancy and the fact that airlines won't let her on a plane now anyway per policy.

It seems they will have a coming-and-going relationship for the next few years until my friend has his citizenship, and can begin the process to bring his wife here. I say 'wife', as they are not yet married because they were advised to NOT do so for the time being, as maybe this would jeopardize her getting a tourist visa again next year to come visit.

When they do marry next year, he may also apply for the I-130 as he can do so as a Perm. Resident but as you say there is a multi-year waiting list or that anyway.

If she applies for an extension, she will still be in status while it is being adjudicated. If it not approved, she will be told to leave the US. If she makes a timely departure, she will not charged with any overstay time.

Make sure she gets a passport for the baby.