OK. So the hole runs part or all of the way through the part and is not for a dead or live center. If your chuck has exactly 0.010" runout, and you drill a hole through the part, take the part out if the chuck, and put it back in at random, the hole will have between 0.000" and 0.010" runout, depending upon how close to exactly as it was in the chuck when you drilled the hole the part is when you put it back in. And assuming that the part is perfectly round.
If you turn the OD and drill the hole in one sitting, the hole and the OD should be concentric. For the time being until you sort the 3-jaw problem out, I would suggest that you use a 4-jaw.
EDIT:
If you hold the bushing in the chuck, however you enlarge the bore the ID is going to be off-center by the chuck runout.
To begin to fix the problem, I would start by removing the chuck and checking for runout on the spindle register. There shouldn't be any to speak of. If there is, and if it is consistent, either use a 4-jaw or a much more expensive adjustable 3-jaw. So long as it makes up on the spindle the same every time, the adjustable 3-jaw will compensate for spindle runout so long as the spindle isn't bent.
Another solution if the spindle taper has no significant runout and the bushing is small enough would be to use a collet.
