I am having a difficult time finding a tapered nylon (i think) bushing to fit into a 1" tube (id .870) which extends beyond the tube about .125. This enables the tube to telescope into another larger tube and along with a tube reducer cap, eliminates the rattle.
Does anyone know of this part or the proper name for this part?

It sounds like it could be a type of compression fitting for tube and pipe. Look for plastic compression tube fittings.

For plastic try Nylok