  We investigate bounds on the strength of the primordial magnetic field (PMF)
from the cosmic microwave background (CMB) bispectra of the intensity
(temperature) modes induced from the auto- and cross-correlated bispectra of
the scalar and tensor components of the PMF anisotropic stress. At first, we
construct a general formula for the CMB intensity and polarization bispectra
from PMFs composed of any type of perturbation. Then we derive an approximate
expression which traces the exact shape of the CMB bispectrum in order to
reduce the computation time with respect to a large number of the multipole
configurations, and also show that the non-Gaussian structure coming from PMFs
is classified as the local-type configuration. Computing the signal-to-noise
ratio on the basis of the approximate formula with the information of the
instrumental noises and resolutions, we find expected upper bounds on the
magnetic field strength, when the magnetic spectrum is nearly scale invariant
($n_B = -2.9$), smoothed on $1 {\rm Mpc}$ scale at 95% confidence level from
the WMAP and PLANCK experiments as $B_{1 \rm Mpc} < 4.0 - 6.7 {\rm nG}$ and
$3.8 - 6.5 {\rm nG}$, respectively, depending on the energy scale of the
magnetic field production from $10^{14} {\rm GeV}$ to $10^3 {\rm GeV}$. Our new
consequences imply slight overestimations by the previous rough discussions.
