Very recently, by applying the so called Beta integral method to the well-known hypergeometric identities due to Bailey and Ramanujan, Choi et al. [extit{Reduction formula for Srivastava's triple hypergeometric series $F^{(3)}[x, y, z]$}, Kyungpook Math. J. {55} (2015), 439--447] have obtained three interesting reduction formulas for the Srivastava's triple hypergeometric series $F^{(3)}[x,y,z]$. The aim of this paper is to provide three unified reduction formulas for the Srivastava's triple hypergeometric series $F^{(3)}[x, y, z]$ from which as many as reduction formulas desired (including those obtained by { Choi et al}.) can be deduced. In the end, three unified relationships between Srivastava's triple hypergeometric series and Kampé de Fériet function have also been given.