A coring bit for extracting a sample of subterranean formation material from a well bore may include a bit body having a bit face and an inner surface defining a substantially cylindrical cavity of the bit body. A first portion of the inner surface may be configured to surround a core catcher. The coring bit may include a face discharge channel inlet formed in the inner surface of the bit body longitudinally at or above the first portion of the inner surface. The coring bit may also include a face discharge channel extending through the bit body from the face discharge channel inlet to the bit face. A tubular body having a core catcher may be disposed in the coring bit to form a coring tool. Methods of forming such bit bodies may include forming an inlet for a face discharge channel in the inner surface of the bit body at a location longitudinally at or above the first portion of the inner surface and forming a face discharge channel extending from the inlet to the bit face.