  In extensions of general relativity and in theories aiming at unifying
gravity with the forces of the Standard Model, the value of the "fundamental
constants" is often determined by the vacuum expectation value of new fields,
which may thus change in different backgrounds. Variations of fundamental
constants with respect to the values measured today in laboratories on Earth
are expected to be more evident on cosmological timescales and/or in strong
gravitational fields. In this paper, I show that the analysis of the K$\alpha$
iron line observed in the X-ray spectrum of black holes can potentially be used
to probe the fine structure constant $\alpha$ in gravitational potentials
relative to Earth of $\Delta \phi \approx 0.1$. At present, systematic effects
not fully under control prevent to get robust and stringent bounds on possible
variations of the value of $\alpha$ with this technique, but the fact that
current data can be fitted with models based on standard physics already rules
out variations of the fine structure constant larger than some percent.
