You can say “Les directeurs s’approchent de moi”, because we say “s’approcher de quelque chose” not “approcher quelque chose”.

Well. You probably want “les directeurs s’approchent de moi”.

“Les directeurs m’approchent” is correct, but has another meaning, “the managers take contact with me”.

Approcher can be used both transitively, intransitively, and as a pronominal verb (s’approcher). All three possibilities produce a slightly different meaning. I’ll give some examples bellow. The translation you provided “Les directeurs m’approchent” corresponds to a transitive use of the verb and is probably the best translation for your original sentence—however it may sound less natural in French.

Pronominal use:

Il s’approche de la vérité.
Les directeurs s’approchent de moi.

Meaning approximatively:

• He is getting close to the truth. (Actively)
• The managers are coming close to me.

Intransitive use:

Il approche de la vérité.
Je voulais empêcher qu’il n’approche de moi.
(Les directeurs approchent de moi.)

Approximatively:

• He gets close to the truth. (It happens)
• I wanted to prevent him to get close to me. (I don’t want it to happen.)
• The managers get close to me. (It happens)

Remark: The last sentence is inside brackets because it would be hard to find a context in which it is appropriate. Such a phrasing would mean that the managers’ intentions are purely irrelevant, they just happen to come close.
(I’m not sure the translation reflects that.)

Transitively:

Il approche le ministre.
Les directeurs m’approchent.

Approximatively:

• He is approaching the minister. (That is, he most probably wants to speak.)
• The managers are approaching me.

That’s your original sentence. Notice however that, in everyday life, French speakers would almost always use the pronominal form instead to describe a scene, unless there is something inappropriate with someone approaching someone else.

Oh, and to make things even more complex, there is another use case for the transitive form:

Il approche l’enfant de la table.

He brings the child closer to the table.