A modification is considered of the well-known Fredrickson-Andersen facilitated kinetic Ising model for studying the glass transition in supercooled liquids. In this modification, which we call the cluster facilitated kinetic Ising model (CluFKI) model, the flip of a spin between the liquid-like regions represented by A or up states and the solid-like regions represented by B or down states can occur (i.e. is facilitated) only if in the cluster containing the spin and those adjacent to it there are at least n spins in the A state. As a result, the transition from a solid-like to a liquid-like region requires more free volume than the reverse transition, which is physically very reasonable. Computer calculations for the n = 1 CluFKI model, as well as a qualitative analysis, show that as the temperature T of the system decreases, and so the parameter x = E/T increases, the fraction fA of spins in state A when the system is in a metastable steady state decreases until it reaches zero for a critical value xc. Correspondingly, the mean relaxation time 〈τ〉 of the system's memory function increases rapidly, and 〈τ〉→∞ as x→xc. This behaviour is similar to that for the α-relaxation in supercooled liquids as the glass transition temperature is approached.