This work is an extension of very recently developed decomposition method for matrices. That method has been called "Tridiagonal Matrix Enhanced Multivariance Product Representation, or briefly, TMEMPR. Here, in this work our ultimate goal has been taken as the decomposition of a univariate linear integral operator. Instead of this task we focus on a bivariate function since the kernel of such an operator is a bivariate function. After having a well developed theory it is just a matter of simple translation what we are going to obtain into linear integral operator's decomposition. The main skeleton of the issue has been constructed in this presentation.