Shakeel -

I don't completely understand your question.  I think you can generally expect a positive intracluster correlation coefficient in practice, because clusters that would naturally occur in a survey would likely have a positive covariance between elements in the cluster.  (I think K.R.W. Brewer (2002), Combined Survey Sampling Inference, Arnold, mid-page 111 perhaps indicates we can expect this in an example involving large retail stores as aggregations of smaller ones.)  If you take a simple random sample instead of a cluster sample, then variance among the lower level units (elements in cluster sampling) is smaller than for cluster sampling, as cluster sampling is generally done for purposes of convenience. 

See pages 241 and 242 in W. G. Cochran(1977), Sampling Techniques, 3rd edition, Wiley.  There Cochran shows that the variance of the estimated mean per element in cluster sampling is the variance of the estimated mean per element for an equal number of elements drawn randomly, multiplied by  [1 + (M - 1)rho], where rho is the intracluster correlation coefficient, and M is the number of elements per cluster, when we are talking about clusters of equal size in single-stage cluster sampling.  Perhaps that is an example of what you were referring to when you asked about eliminating rho by using random sampling??

But you said "Does this correlation become zero if we select a random sample from the selected clusters," and that would be "no," because you cannot have selected clusters first.

Well, those are some thoughts which may or may not be appropriate.

Cheers - Jim

PS - To clarify the last part of my answer above, you can take a cluster sample by some design, and then do second stage sampling within each cluster by another design, but the fact that cluster sampling is involved at all will, I think, generally mean you will have a positive intracluster correlation with which you must contend.