Game of Thrones can help you remember the correct phrase.
@Usedtobeme on Twitter asked, "Is it deep-seated or deep-seeded?”
The correct phrase is deep-seated.
Seat can mean a location, as in the county seat, or a body part that is the center of some emotion or function. For this definition, Dictionary.com and the American Heritage Dictionary both use this example sentence: The heart is the seat of passion.
We don't often use the word seat that way anymore, so it's easy to see why people get confused about deep-seated. When you're thinking of something deeply felt or buried, it's not far fetched to think of a seed buried in the dirt. Nevertheless, deep-seated is the right choice. The Oxford English Dictionary defines deep-seated as "having its seat far beneath the surface.”
Instead of a seed buried in the dirt, think of a big, deep seat buried in your backyard—maybe even a throne like the one in Game of Thrones. That’s a memorable image.