Recently, Castelani and Guili extended to equilibrium problems a result on existence of solutions of quasi-monotone Stampacchia variational inequalities, established by Ausel and Hadjisavvas. The proof of this result is based upon KKM techniques. In this paper we define the so-called quasi-equilibrium problem, which is an equilibrium problem with a nonconstant constraint set, depending on the  variable of the problem. Our main contributions consist of an existence result, which is established through a combination of fixed point techniques and stability analysis of perturbed equilibrium problems.