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.