You should know that the cat's behavior is more controlled by the sense of smell than the sense of sight.
Interested in Language
I would like to say that whenever our cat sees us moving our mouth he licks his mouth.
How about "Each/every time our cat sees us eating something he licks his mouth"?
" Our cat licks his lips at the sight of anything /that/ we eat".
Do you think the two sentences in inverted commas express the idea stated in the heading?
Whenever our cat sees us eating (anything), he starts licking his chops.
He licks his chops, yes. And he says: "Waow!"
Or licks his lips.
Bear in mind that you seem to be equating "moving your mouth" with "eating". I move my mouth when I talk.