The Morris constant term identity is important due to its equivalence with the well-known Selberg integral. We find a variation of the Morris constant term, denoted $h_n(t)$, in the study of the Ehrhart polynomial $H_n(t)$ of the $n$-th Birkhoff polytope, which consists of all doubly stochastic matrices of order $n$. The constant term $h_n(t)$ corresponds to a particular constant term in the study of $H_n(t)$. We give a characterization of $h_n(t)$ as a polynomial of degree $(n-1)^2$ with additional nice properties involving the Morris constant term. We also construct a recursion of $h_n(t)$ using a similar technique for the proof of the Morris constant term identity by Baldoni-Silva and Vergne, and by Xin. This method enables efficient computation for fixed $n$. Additionally, we introduce a conjecture, which establishes a simple product formula for the characteristic polynomial of an $(n-1) \times (n-1)$ tridiagonal matrix $C$. We have proved this conjecture in a sequel paper. This work is joint with Jiaqiang Hu, Guoce Xin.