  The spacetime geometry around astrophysical black holes is supposed to be
well approximated by the Kerr metric, but deviations from the Kerr solution are
predicted in a number of scenarios involving new physics. Broad iron K$\alpha$
lines are commonly observed in the X-ray spectrum of black holes and originate
by X-ray fluorescence of the inner accretion disk. The profile of the iron line
is sensitively affected by the spacetime geometry in the strong gravity region
and can be used to test the Kerr black hole hypothesis. In this paper, we
extend previous work in the literature. In particular: i) as test-metric, we
employ the parametrization recently proposed by Konoplya, Rezzolla, and
Zhidenko, which has a number of subtle advantages with respect to the existing
approaches; ii) we perform simulations with specific X-ray missions, and we
consider NuSTAR as a prototype of current observational facilities and eXTP as
an example of the next generation of X-ray observatories. We find a significant
difference between the constraining power of NuSTAR and eXTP. With NuSTAR, it
is difficult or impossible to constrain deviations from the Kerr metric. With
eXTP, in most cases we can obtain quite stringent constraints (modulo we have
the correct astrophysical model).
