  In the type-I seesaw mechanism for neutrino masses, there exists a $B-L$
symmetry, whose breaking leads to the lepton number violating mass of the heavy
Majorana neutrinos. This would imply the existence of a new neutral scalar
associated with the $B-L$ symmetry breaking, analogous to the Higgs boson of
the Standard Model. If in such models, the heavy neutrino decays are also
responsible for the observed baryon asymmetry of the universe via the
leptogenesis mechanism, the new seesaw scalar interactions with the heavy
neutrinos will induce additional dilution terms for the heavy neutrino and
lepton number densities. We make a detailed study of this dilution effect on
the lepton asymmetry in three generic classes of seesaw models with TeV-scale
$B-L$ symmetry breaking, namely, in an effective theory framework and in
scenarios with global or local $U(1)_{B-L}$ symmetry. We find that requiring
successful leptogenesis imposes stringent constraints on the mass and couplings
of the new scalar in all three cases, especially when it is lighter than the
heavy neutrinos. We also discuss the implications of these new constraints and
prospects of testing leptogenesis in presence of seesaw scalars at colliders.
