A smooth variety X of dimension n is said to satisfy the diagonal property if there exists a vector bundle ε of rank n on X × Xand a section s of ε such that the image Δ(X) of the diagonal embedding of X into X × Xis the zero scheme of s. A study of varieties satisfying the diagonal property was begun by Pragacz, Srinivas and Pati, in [8]. Even though there are many cases where the answer is affirmative, only in a few examples an explicit description of the vector bundle is known. After an exposition of toric varieties, we discuss this question in the particular case when X is a toric surface, in search for such examples.