A subalgebra B of a Leibniz algebra L is called a weak c-ideal of L if there is a subideal C of L such that (Formula presented.) and (Formula presented.) where (Formula presented.) is the largest ideal of L contained in B. This is analogous to the concept of a weakly c-normal subgroup, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterizations of solvable and supersolvable Leibniz algebras generalizing previous results for Lie algebras. We note that one-dimensional weak c-ideals are c-ideals, and show that a result of Turner classifying Leibniz algebras in which every one-dimensional subalgebra is a c-ideal is false for general Leibniz algebras, but holds for symmetric ones.