This is a technical question depending small losses for example in clean room applications (every particle matters...)
I assume you have tried shielded bearings, where the grease is contained inside covers that prevent its loss? Or possibly you do not have the funds to replace the bearings? Etc. I do not know the size of your pump/motor or whatever the bearing is on. I will tell you I have seen in an industrial setting on what I would call a small pump, they actually put a bag around the bearing housing to catch grease. Of course, the shaft ended at the housing so the bag did not have to come in contact with the rotating assembly. You could probably get more useful answers if you re-ask your question describing the size and physical configuration of the bearings and their housing.
