Abstract Let L ∗ denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L ∗ contains all sufficiently large integers. That is, there is a smallest integer n o such that L ∗ contains all integers greater than n o. However, no upper bound for n o has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for n o. In particular it is shown that L ∗ contain all integers n>5594, with the possible exception of n=6810.