No, at least not most animals. There was a study a while back that showed that animals think by reducing the world to a series of binary choice that they react to in the moment. I imagine it's a lot like when you're playing a sport or video game and things get very intense and fast paced; your inner monologue isn't telling you what your next move will be every second, you're just reacting on instinct. That's probably how animals see the world all the time.
That being said, "animals," is a broad category, and some of them may be capable of creating an abstract narrative for themselves. It was recently discovered that whale songs have a phonetic alphabet, which means their language may be as complex as ours. If that's the case, they may be capable of using that language to build an internal monologue.