The answer is..that depends.

If it's a child, or grandchild then the tax basis can be transferred. If you have a corporation owning the home, rather than an individual, or the family member is another relation than mentioned, then it would trigger a 'tax assessment event' and be reassessed.

I don't know what your current tax basis is, but I'll bet it's more than what you owed on the loan. So they would be better off not transferring the tax basis....

All of this is elements for discussion. I'd call the tax assessor's office to get a clearer picture of your specific situation. Alot depends on how you transfer title and what is recorded in that transfer.