Leaks in a tyre rim most likely result from the type of alloy material used on most cars. Previously, steel rims leaked air through the rim and the tyre. Alloy rims cause a leak through porous holes found in cast-aluminium alloy. The holes can be repaired, but you may wind up replacing the rims or tyres if the leak persists.
Inflate the tyre with the rim leak up to 2.75 BAR (39 PSI).
Remove the rim and place it in a large bucket of water. Note areas in which bubbles form. This indicates the origin of the leak.
Mark the areas with a grease pencil and take the rim out of the water.
Demount the tyre from the rim.
Sand the area with 80-grit sandpaper and brush off any debris or remnants.
Dab the area lightly with a towel and clean the leak area with carburettor cleaner or white spirit.
Cover the area thoroughly with a silicone gasket sealant, which should close the leak.
Let the rim dry for a few hours before remounting it on the tyre.
Refrain from using silicone tub caulking -- it may cause the metal wheel to become corroded.