For a precision hole you really need to ream. HSS twist drills don't really drill round holes if the center web is engaged in the work, they drill somthing more like a Reuleaux triangle. If you don't have a reamer, you can get closer to round and on size by drilling a pilot hole first, to relieve the center area for the drill of the target size.
There are a lot of variables that go into precision hole making, so in general if you want a press fit on a .0937" ball, you would be shooting for a .0927-.0930" hole. One repeatable way of getting there would be to drill with #44 (.086) drill, and then reaming with a .0927" reamer. If you have a drill press with little run out, sufficient spindle speed and good drilling practices (allowing the drill to cut, rather than forcing, etc) you may be able to get a sufficient press fit by drilling with a #44 bit and then following that with a #42, but I doubt it. A #43 drill (.089) with no pre-drilled hole may go over size enough to get the ball pressed in, but that's not very reliable either.
But drilling and reaming .0937 when your ball is .0937 (size on size) will never give you a good secure press fit. Not one that lasts more than a couple open-close cycles.