We consider the following question for the ideals Π _p of absolutely p-summing operators: Is it true that, for given Banach spaces X and Y, the unit ball of the space Π _p(X,Y) is dense, for some natural topology, in the unit ball of the space Π _p(X,Y**) or in the unit ball of the corresponding space Π _p ^dual(Y*,X*):=(U:Y*→X*ιU* ι _XεΠ _p(X,Y**)? As "natural topologies", we consider strong and weak operator topologies, compact-open topology, topology of X × Y*-convergence etc.