It's possible to apply for a Change of Status (change from one nonimmigrant status to another within the US).
Changing into or extending a work status like E-1, E-2, E-3, H-1C, H-1B, H-1B1, H-2A, H-2B, H-3, L-1, O-1, O-2, P-1, P-2, P-3, Q-1, R-1, or TN, can only be done by the employer filing I-129, rather than you filing I-539 (which is the way to change into or extend other nonimmigrant statuses). In the case of H-1b, L-1, O-1, etc., the employer needed to file I-129 to petition for you anyway (regardless of whether you are doing Change of Status in the US or a visa abroad), so they don't need to do any extra work to also request a Change of Status or Extension of Status. But in the case of TN, a petition is normally not needed -- you normally only need an employer offer letter to gain entry into TN status (for Canadians) or to get a TN visa (for Mexicans). But in the case where you want to do Change of Status into TN or Extension of Status for TN inside the US, the employer has to do the extra work of filing I-129 (with a $460 fee). If the employer is not willing to file I-129, your only option is to get a TN visa abroad.
I am not sure what you mean by "My potential employer has started the process to request a TN visa". If you mean that they have filed I-129, then they should have also requested a Change of Status on it too.
And as phoog mentioned, if you leave the US, you will need to get a TN visa to return to the US anyway. Status is only for a particular stay; you need a visa to enter the US. And as phoog also mentioned, Change of Status takes several months, so it is unlikely for you to be approved by November if your employer files around now. It is much faster to leave and get a visa.