Your question is not clear
Are you trying to draw circle in a drawing or part
Do you have any picture to post here?
Place your sketch for the new circle on the bottom plane of the "bored" area. You can then convert the bottom of the "Bore" to a construction line circle and then assign a concentric relation to it. If it is not to be concentric, you should be able to dimension to the converted center.
There are other ways (relating to an axis, etc).
Try this approach:
Sketch the circle at the bottom of the bore off center a good bit.
You know, off in space where you don't want it.
Click on the center of the circle you sketch and move it to hover over the center of the bore hole (not the edges), and you should be able to constrain the centers Vertically or Horizontally or Co-incident. You can also control an offset by dimensioning from by clicking the Edges of each