Motivated by the lack of direct evidence with inelastic neutron scattering of the well documented bound state of Heisenberg ferromagnets, we use the time-dependent thermal density matrix renormalization group algorithm to study the temperature dependence of the dynamical spin structure factor of Heisenberg ferromagnetic spin chains. For spin 1/2, we show that the bound state appears as a well defined excitation with significant spectral weight in the temperature range $J/12\ensuremath{\lesssim}T\ensuremath{\lesssim}J/3$, pointing to the possibility of detecting it with inelastic neutron scattering near $k=\ensuremath{\pi}$ provided the temperature is neither too low nor too high---at low temperature, the spectral weight only grows as
T3/2 and, at high temperature, the bound state peak merges with the two-magnon continuum. For spin 1, the situation is more subtle because the bound state with two neighboring spin flips competes with an antibound state with two spin flips on the same site. As a consequence, the relative spectral weight of the bound state is smaller than for spin 1/2 and a weak resonance due to the antibound state appears in the continuum. A clearer signature of the bound state (antibound state) can be obtained if a negative (positive) biquadratic interaction is present.