Need help with relative movement between object and player

I have a portrait of an old woman whose eyes I want to follow the player in a very creepy fashion. The “eyes” themselves are two glowing particle objects and are only meant to represent the iris, not the whole eye. All I need them to do is appropriately translate downwards as the player approaches along with moving left or right relative with the players movements.

Vector3.MoveTowards almost does the trick if I force the z axis to remain 0, but the eyes tend to move up or down more than they should for a convincing effect. I’ve also tried using sphere objects that face the player but unfortunately the effect looks really cheesy because it makes the painting look like it has bug eyes.

Any ideas?

Use your spheres facing the player code, but place a particle below the eye gamobject in the hierarchy, so it rotates with its parent and remove the bug eye sphere mesh, leaving only the particle visible, but with the wanted look functionality