  We examine the level of enhancement that can be achieved in the ZZ and
\gamma\gamma channels for a two-Higgs-doublet model Higgs boson (either the
light h or the heavy H) with mass near 125 GeV after imposing all constraints
from LEP data, B physics, precision electroweak data, vacuum stability,
unitarity and perturbativity. The latter constraints restrict substantially the
possibilities for enhancing the gg -> h -> \gamma\gamma or gg -> H ->
\gamma\gamma signal relative to that for the SM Higgs, hSM. Further, we find
that a significant enhancement of the gg -> h -> \gamma\gamma or gg -> H ->
\gamma\gamma signal in Type II models is possible only if the gg -> h -> ZZ or
gg -> H -> ZZ mode is even more enhanced, a situation disfavored by current
data. In contrast, in the Type I model one can achieve enhanced rates in the
\gamma\gamma final state for the h while having the ZZ mode at or below the SM
rate - the largest [gg -> h -> \gamma\gamma]/[gg -> hSM -> \gamma\gamma] ratio
found is of order ~1.3 when the two Higgs doublet vacuum expectation ratio is
tan\beta = 4 or 20 and the charged Higgs boson has its minimal LEP-allowed
value of m_{H^\pm} = 90 GeV.
