The linear stability of a class of force-free equilibria in cylindrical geometry is investigated. The class consists of cylindrically symmetric force-free equilibria for which the ratio between the parallel current density and the magnetic field is a step function of the radius. It is suggested that plasmas in reversed field pinches could be roughly represented by such equilibria as a consequence of a small departure from an initial force-free state with constant , the latter being reached after a relaxation process according to the classical theory proposed by Taylor [Phys. Rev. Lett. 33 , 1139 (1974)]. A fully analytical derivation of the tearing stability parameter for such class of equilibria is given. It is then shown with one explicit example how the presence of a downward step of relatively small height can destabilize the innermost resonant mode, which would otherwise be stable if were constant. A possible implication of this mechanism for the formation of cyclic quasisingle helicity states observed in reversed field pinches is proposed. Considerations on the ideal stability of the class of equilibria under investigation are also given.