  In a previous paper, one of us has showed that, at least in some cases, the
Kerr-nature of astrophysical black hole candidates is extremely difficult to
test and current techniques, even in presence of excellent data not available
today, cannot distinguish a Kerr black hole from a Bardeen one, despite the
substantial difference of the two backgrounds. In this paper, we investigate if
the detection of the "shadow" of nearby super-massive black hole candidates by
near future mm/sub-mm very long baseline interferometry experiments can do the
job. More specifically, we consider the measurement of the Kerr spin parameter
of the Bardeen and Hayward regular black holes from their shadow, and we then
compare the result with the estimate inferred from the K$\alpha$ iron line and
from the frequency of the innermost stable circular orbit. For non-rotating
black holes, the shadow approach provides different values, and therefore the
Kerr black hole hypothesis can potentially be tested. For near extremal
objects, all the approaches give quite similar results, and therefore it is not
possible to constrain deviations from the Kerr solution. The present work
confirms that it is definitively challenging to test this kind of metrics, even
with future facilities. However, the detection of a source that looks like a
fast-rotating Kerr black hole can put meaningful constraints on the nature of
the compact object.
