There's a pervasive misunderstanding that someone in TN status cannot pursue a green card and must be in H-1B status to do. This is just not true.

You can technically apply for a green card but border guards can and do use it as evidence that you don’t intend to stay “temporarily” and deny your TN as a result.

They can reevaluate your TN status at every crossing, and there are many stories of this happening to people returning from holiday when I was at microsoft. The TN is efficient to get but has its downsides.

For anyone curious like I was, Trade NAFTA Visa

https://en.wikipedia.org/wiki/TN_status

That's correct so when pursuing a green card while in TN status, the applicant should only travel internationally after commencing the green card process on an advance parole travel document (not on a TN) or get his or her green card through a process abroad, called immigrant visa processing. An applicant for a green card in TN status should never seek admission in TN status if he or she is the beneficiary of a pending or approved I-140 petition unless he or she is immigrant visa processing and only then after consulting with an immigration attorney to understand the issues, process, and risks of doing so.