The chimney is brick - it is unlined and used only for the furnace and hot water heater flues. The chase is really just the framing but probably 4 inches or so from the brick on three sides and about 8-10 inches on the back side. I was "thinking" about putting the conduit up the back side as far from the chimney as possible. The main sanitary line for the second floor and the water lines for the second floor are already in that space but I wanted to bring the conduit from the basement up through the second floor into the attic. I checked the national code book and I couldn't find anything about NOT putting the conduit up through the chase but it did say to make sure the wires and conduit are rated for temp, the temp isn't too high, and you are careful to seal off the space around the conduit so that you prevent open access from basement to attic for fire spreading reasons.
I just got another email from someone reading this posting and they were pretty critical about not doing this. I'm just looking to make a good, sound, safe decision.
THANKS for your help!