  We consider the evidence for very hard low energy spectra during the prompt
phase of Gamma-Ray Bursts (GRB). In particular we examine the spectral
evolution of GRB 980306 together with the detailed analysis of some other
bursts already presented in the literature (GRB 911118, GRB 910807, GRB 910927
and GRB 970111), and check for the significance of their hardness (i.e.
extremely steep spectral slopes below the $EF_{E}$ peak) by applying different
tests. These bursts, detected by the Burst And Transient Source Experiment
(BATSE) in the $\sim 30$ keV -- 2 MeV energy range, are sufficiently bright to
allow time resolved spectral studies on time intervals of the order of tenths
of a second. We discuss the hard spectra of these bursts and their evolution in
the context of several non--thermal emission models, which all appear
inadequate to account for these cases. The extremely hard spectra, which are
detected in the early part of the BATSE light curve, are also compared with a
black body spectral model: the resulting fits are remarkably good, except for
an excess at high energies (in several cases) which could be simply accounted
for by the presence of a supra--thermal component. The findings on the possible
thermal character of the evolving spectrum and the implications on the GRB
physical scenario are considered in the frameworks of photospheric models for a
fireball which is becoming optically thin, and of Compton drag models, in which
the fireball boosts "ambient" seed photons by its own bulk motion. Both models,
according to simple estimates, appear to be qualitatively and quantitatively
consistent with the found spectral characteristics, although their possible
caveats are discussed.
