Modal operators which correspond to impossibility and non-necessity are investigated in systems analogous to the modal logic $K$ which are based on the Heyting propositional calculus. Soundness and completeness are proved with respect to Kripke-style models with two accessibility relations, one intuitionistic and the other modal. A system where impossibility is equivalent to intuitionistic negation is also proved sound and complete with respect to specific classes of models with two relations. It is shown how the holding of formulae characteristic for this system is equivalent to conditions for the relations of the models.