An invoicing arrangement generally describes self-employment or independent contracting, and self-employment in the United States is not permitted under TN. The category is built around prearranged employment with a United States employer, so the payment structure is not a detail that can be adjusted after entry.
The relationship has to be employment before anything else is assessed
Look at what the arrangement actually creates rather than at what it is called. If the American company directs the work, sets the hours, supplies the assignments and pays through payroll, that points toward employment. If you invoice monthly, carry your own insurance, choose your own methods and could send a substitute, that points toward an independent business, and an independent business run in the United States is outside what this category contemplates. The practical step is to ask the company directly whether it is willing to engage you as an employee, because a company that has budgeted for a contractor may simply not be. Where the answer is no, that is useful information early rather than a discovery made at a port of entry, and a licensed adviser can then look at whether any other route fits the commercial reality both sides actually want.
Hypothetical example: Mara, a marine-cartography analyst, is offered a job called consultant but learns that the company expects invoices from her own entity. TN requires a real employer-led role in a listed professional category; a self-directed U.S. business arrangement cannot be converted into eligibility by changing the invoice label. The first review should produce a yes-or-no issue list, identify the record that resolves each issue, and state what must change before a filing or travel plan is chosen. A job description, contract, ownership record, or credential may settle one point while leaving another open, so conclusions should name the document relied upon rather than rest on a general impression.