The original question was more general than "is AES circular secure", since AES was cited as an example only.
You are asking for a circular secure scheme.
This is not as simple as it seems, because the security games that we use to prove security generally go (roughly) like this :
- An oracle generate the keys;
- The adversary chooses two messages $m_0$ and $m_1$, and send them to the oracle;
- The oracle returns a ciphertext $c$;
- The adversary tries to guess if $c$ encrypts $m_0$ or $m_1$.
And the scheme is considered secure if the adversary's advantage in step 4. is negligible.
Therefore, those security games do not cover the case "the adversary chooses one of the messages to be the secret key", because if the adversary could do that with non-negligible probability, they would already be able to win the game.
As a consequence, it may happen that a scheme is secure in normal usage, but insecure if one encrypts the own key. Indeed, researches managed to construct schemes that are CPA secure, but not circular secure.
However, I have read sometimes (even in scientific papers) that not being circular secure is believed to be an anomaly, in other words, in general, people believe that most of the schemes are circular secure. Moreover, this assumption (or a weak circular-security assumption) is used very often to construct advanced schemes like homomorphic encryption ones.
You may want to read this nice post by Matthew Green, which cites papers about this subject so that you can read them if you want more technical information.