In short, no.
In full, anyone can become an ATG but I don't think Narine is more likely to than, say, Ravi Ashwin.
Look WW, Narine has the potential to be good - very, very good. But any discussion of him becoming a Test ATG is extremely premature. I mean, right now I'd have him on par with Tim May in the pantheon of spinners, and that's being generous. I suspect at the end of his career he'll be remembered as a lot better than Tim May, but I genuinely believe it's only a matter of time before he is found out in Test cricket and becomes limited to being a highly effective option in the shorter (especially shortest) format of the game. Like a certain Sri Lankan who is comparable to Narine in almost every way bar nationality.
And thinking through this logically, as a mystery spinner one would expect Narine to be at his most successful towards the start of his career, like an Iverson, Gleeson, Ashwin or Mendis. Gleeson, in particular, is the archetype of the career path we'd expect a mystery spinner to take. Narine hasn't started off nearly as brightly as any of those men, and once people work him out in Tests (which won't take long compared to T20), he's going to be distinctly mediocre.
In T20 you can make a career out of bowling flat and beating batsmen off the pitch. In Tests you have to beat them in the air. Mendis found that out, Narine discovered that in his early days of Tests, and Ashwin has adapted his bowling to try and deal with that. Narine will have to become a drastically different bowler to the one he is now if he's to have a successful Test career. With all the T20 money being thrown at him, I don't think that's likely. Him having played 125 T20s and 13 FC games is a pretty damning statistic tbh.
He's a T20 ATG already, but IMO there is little chance of him being one in Tests.