It's the most standard Mahayana teaching. The references are too numerous to cite, almost every other text has statements to this effect. However, it should never say "You are already Enlightened" (or anything naive like that), the phrases are more nuanced and the meaning is more subtle.
For example, it may say, "Our nature is fundamentally pure" or "Your original nature is no different from that of the Buddhas" etc. - in an attempt to turn our attention to the existential conflict between our dualistic mind of "this is wrong and I need to be something else" and the perfect suchness of Nirvana in the here-and-now.
Now, if you speak with an actual Zen (or another Mahayana) teacher, they will be quick to point out that even though our primordial nature is indeed Nirvana, the habits of craving and clinging are extremely difficult to overcome on the spot, which is why we must cultivate Sila/Prajna/Samadhi.
So in some sense the teaching has two sides and which one your teacher will drum depends on which way you lean in your particular state of confusion. If you are inclined to be complacent they will drum the Path and if you are obsessed with the goal, they will drum Buddha-Nature. It's kinda frustrating because you end up being wrong no matter which position you take. It's only when you mature beyond positions is when this contradiction resolves.