I've never seen it mentioned, specifically, but we can get close to a minimum age.
July 7, X777 is when the dragons disappeared. Since Natsu was trained by Igneel, he must have been born before this date, presumably several years before to allow time for training and such. If Natsu had been born the day before, he would be about 14.
During the Fighting Festival arc, the Battle of Fairy Tail specifically, Natsu, Gajeel, and Makarov are trapped by Freed's enchantment, suggesting that Natsu must be older than 80 at this point.
The Edolas arc sees the return of Gildarts after 3 year absence, suggesting the Battle of Fairy Tail was less than 3 years prior to this. Since there is no minimum time that passed, there is not much help to us here.
In the X971 arc we know 7 years have passed, so Natsu must be at least 87 by this point.
Beyond that, I'm not sure how much time is passing between each event.
I do have a theory that would make sense of why Natsu is so old when physically he looks young:
Natsu and the other Dragon Slayers are actually their respective dragons. For example, Natsu is actually Igneel.