Does the solubility of reactants, in the reaction solvent, affect the rate of the reaction?

The reactants need to both dissolve, at least partially, in order to interact. If one reactant is only partially soluble, the rate of the reaction will depend on the mechanism for that specific reaction.
For example, the rate determining step of a unimolecular substitution reaction involves only one species, so the rate depends on the concentration (only considering the amount that has dissolved and is available to react) of only one of the reactants. For a bimolecular substitution, the concentrations of both reactants will determine the rate.