@JTT,
I don't know that we particularly use "eyeballing"......hmmm.....
I think, if we did, that I would take it to mean "getting a look at", it wouldn't have the somewhat threatening sense of "giving the hairy eyeball".
Here's an American type definitions of eyeball used as a verb:
tr.v. eye·balled, eye·ball·ing, eye·balls Informal
1. To look over carefully; scrutinize.
2. To measure or estimate roughly by sight: eyeballed the area of the wall that needed paint