We theoretically discuss the group delays induced by the transverse-magnetic polarized light penetrating a Weyl semimetal (WSM) film. It is shown that the transmitted group delay τ_ps emerges because of the nonzero off-diagonal components of the dielectric constant tensor. In particular, the transmitted group delays τ_pp and τ_ps are enlarged significantly at the epsilon-near-zero (ENZ) frequency, where steep phase changes occur due to the sharp changes of the transmission coefficients | t_pp | and | t_ps | . More importantly, the tunable ENZ effect can be effectively obtained with Fermi energy, so the giant enhanced group delays at various frequencies can be readily realized. The enhanced group delays also show the adjustability of WSM thickness, incident angle, and Weyl node separation. Our findings supply easy and available methods to realize the adjustable enhanced group delays with a WSM.