I had this issue with a walnut Garand stock. The basic technique Brownells recommended at the time was to make a slurry of whiting (agricultural lime works) and tri-chlor and apply it to the surface, and the vapor would penetrate and the oil was leeched out into the dust. After the solvent was evaporated, putting it in an oven (150°F) caused still more to wick out.
These days you'd have to settle for some other solvent. I suppose mineral spirits would work, too, but it would be slow. I think I would try liquid (not paste) paint stripper. It would be faster and would tend to remove the old finish with it. You'd want to use it outdoors, though. I also recommend you buy a 5 lb bag of ag lime, as it seems to me it took about 20 applications to get oil to stop coming out. I don't expect it was gone 100% at the core from mne, but it was at least gone far enough from the surface that I could put a new finish on and have it dry.