Eyebolt where nut is another eyebolt?
Yes, there are available parts like you ask about. Here is a picture of one type:



If you do a Google search with "eye nut" you can find more pictures, descriptions and sources of supply. 

You may want to consider using two of these and then run a threaded rod through the tubing. In addition to the above type eyes (fixing or rigging hooks) you would use two nuts on each side of the threaded rod. One from each side to secure the threaded rod to the metal tubing and the second one threaded on before the eye nut. The second nut would be double nutted with the eye nut (i.e. tightened against the eye nut to secure the position of the eye in the proper orientation with respect to the eye on the other end of the tubing.

You could simplify the above suggestion by cutting the threaded rod just the right length and using just the eye nuts to secure the rod to the tube. In this case you would want to use a thread locker (such as LocTite) to lock the threads when the two eyes are in the desired orientation with each other.