Board Thread:General Mod Discussion/@comment-33185350-20171014150536/@comment-33185350-20171017133812

Yes it would be in my opinion. In The Hobbit, Tolkien writes like he hadn’t come up with Gollum’s backstory yet. All he is is one of the nameless things at the bottom of the world - “Deep down here by the dark water lived old Gollum, a small slimy creature. I don’t know where he came from, nor who or what he was. He was Gollum-as dark as darkness, except for two big round pale eyes in his thin face.”

Also you will see from the quote in the original post it was Gollum’s intention to get deep down to the Roots of the Mountains. So I think it’s reasonable that we take his cavern to be there.