  We study whether a 125 GeV standard model-like Higgs boson can be
accommodated within the scale-invariant NMSSM in a way that is natural in all
respects, i.e., not only is the stop mass and hence its loop contribution to
Higgs mass of natural size, but we do not allow significant tuning of NMSSM
parameters as well. We pursue as much as possible an analytic approach which
gives clear insights on various ways to accommodate such a Higgs mass, while
conducting complementary numerical analyses. We consider both scenarios with
singlet-like state being heavier and lighter than SM-like Higgs. With A-terms
being small, we find for the NMSSM to be perturbative up to GUT scale, it is
not possible to get 125 GeV Higgs mass, which is true even if we tune
parameters of NMSSM. If we allow some of the couplings to become
non-perturbative below the GUT scale, then the non-tuned option implies that
the singlet self-coupling, kappa, is larger than the singlet-Higgs coupling,
lambda, which itself is order 1. This leads to a Landau pole for these
couplings close to the weak scale, in particular below ~10^4 TeV. In both the
perturbative and non-perturbative NMSSM, allowing large A_lambda, A_kappa gives
"more room" to accommodate a 125 GeV Higgs, but a tuning of these A-terms may
be needed. In our analysis we also conduct a careful study of the constraints
on the parameter space from requiring global stability of the desired vacuum
fitting a 125 GeV Higgs, which is complementary to existing literature. In
particular, as the singlet-Higgs coupling lambda increases, vacuum stability
becomes more serious of an issue.
