Polyurethane glues bond to both rubber and wood, among other materials, according to the Adhesive and Sealant Council. This glue can help hold both rubber and wood together.
Polyurethane glue is a type of polymer adhesive that is relatively new on the market. This adhesive is available in one- or two-step products. It works best on slightly moist surfaces, which can be achieved by wiping down the wood and rubber with a damp cloth. Both surfaces need to be clamped together to form a tight bond.
During the drying process, a brown foam may leak out between the two materials. This is normal, and the excess can be cut away after it dries.