We calculate the shear viscosity η in the quark-gluon plasma (QGP) phase within a virial expansion approach with particular interest in the ratio of η to the entropy density s, i.e. η/s. The virial expansion approach allows us to include the interactions between the partons in the deconfined phase and to evaluate the corrections to a single-particle partition function. In the latter approach we start with an effective interaction with parameters fixed to reproduce thermodynamical quantities of

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... CD such as energy and/or entropy density. We also directly extract the effective coupling _ V for the determination of η. Our numerical results give a ratio η/s≈ 0.097 at the critical temperature T_ c, which is very close to the theoretical bound of 1/(4π). Furthermore, for temperatures T≤ 1.8 T_ c the ratio η/s is in the range of the present experimental estimates 0.1-0.3 at RHIC. When combining our results for η/s in the deconfined phase with those from chiral perturbation theory or the resonance gas model in the confined phase we observe a pronounced minimum of η/s close to the critical temperature T_ c.