Indeed there is such a form but it is also quoted speech. Do you hear sometimes when people end their statement in dictionary form? 아프다 for example. Basically you treat the whole thing as a noun phrase so as if you reported it as a noun you use 이라고 but 이 of course is dropped off right after 다.
From time to time i see that it is really similar to quoted speech so i think it is just a bit redundant but basically you just think of it as if you reported exactly when you hear others end their statement in dictionary form, literally