Hi,

The whites in eyes of my bengal, Zack, are quite red and bloodshot and have been for a couple weeks now.

They aren't running excessively but I notice he does have a daily buildup of what I might call "sleepies" in the corners of his eyes from dried tears or whatever it may be. Other than that, his eyes seem fine and he doesn't seem sick in any way.

I'm wondering if this would be considered normal or something I should maybe be worried about?

How old is zack?
How often are you seeing his discharge build-up?
Just after a sleep?
There may be more discharge than you are aware of due to him wiping it away often himself while he's awake.

Conjunctivitis is a possibility but there are several other medical conditions (some more serious) that can cause bloodshot eyes as a symptom.
Its not normal and should be checked out.

An environmental irritant could be another cause for this... but IMO you should get the vet to rule out anything more serious before looking for an allergic reaction.

take a pic to show us ?