  In a previous paper, one of us has described a code to compute the thermal
spectrum of geometrically thin and optically thick accretion disks around
generic stationary and axisymmetric black holes, which are not necessarily of
the Kerr type. As the structure of the accretion disk and the propagation of
electromagnetic radiation from the disk to the distant observer depend on the
background metric, the analysis of the thermal spectrum of thin disks can be
used to test the actual nature of black hole candidates. In this paper, we
consider the 10 stellar-mass black hole candidates for which the spin parameter
has been already estimated from the analysis of the disk's thermal spectrum and
under the assumption of the Kerr background, and we translate the measurements
reported in the literature into constraints on the spin parameter--deformation
parameter plane. The analysis of the disk's thermal spectrum can be used to
estimate only one parameter of the geometry close to the compact object, and
therefore it is not possible to get independent measurements of both the spin
and the deformation parameters. The constraints obtained here will be used in
combination with other measurements in future work, with the final goal to
break the degeneracy between the spin and possible deviations from the Kerr
solution and thus test the Kerr black hole hypothesis.
