I would not agree to pay them anything, especially if you don;t have a written contract.
If they report a 1099-NEC instead of a W-2, the first thing you do is file form SS-8. This is a request to have the IRS determine your status. If you have pay stubs showing withholding of taxes, especially SS taxes, you can include that as proof, along with any other information suggested in the instructions.
https://www.irs.gov/forms-pubs/about-form-ss-8
You might remind the company about the SS-8 form, and the heavy penalties that companies pay if they are found to have misclassified employees.
As regards to your tax return, that is a little more complicated. If they withheld social security and medicare tax from your paychecks, it will be almost impossible for them to report your pay on a 1099-NEC and still balance their books. If that does happen, form 8919 is probably not correct, because it will tax you for half the social security that was already withheld. I would see an accountant for help (and not a storefront). Probably you would use a substitute W-2 form instead, but this will also cause an investigation and likely delay any refund.
However, it sounds like they might only want to issue a 1099 for the money paid for the training, as if it was paid to you. This would also be improper, and would not make any sense. If you are a regular employee (a common law employee) then the company can deduct your training as a business expense. If they report it as 1099 compensation, it still is a business expense to them (they don't get a better deduction) but it hurts you, so the only reason to do that is spite. It would not be taxable to you in that case. I think your best option in that case would be to file your return by mail, not including the training money. Instead, print your return and mail it in. Attach a copy of the 1099 and a written letter of explanation that also says you have filed an SS-8. (And you will include the 1099 and the explanation as part of your case when you file the SS-8).
It is ultimately unclear to me whether they want to 1099 you for the wages, the training, or everything. Best to wait until you actually get the documents and then we can talk about how to handle it.
