It has been pointed out that the physical states in linearized quantum gravity are required to be invariant under the continuous isometries of the background spacetime if the Cauchy surfaces are compact. This requirement would appear to allow only the vacuum state as the physical state in linearized quantum gravity in de Sitter spacetime. The first step toward resolving this apparent paradox is to construct a new Hilbert space of de Sitter-invariant states. In this article an approach to this task is presented. First de Sitter-invariant states with infinite norm are constructed by smearing the states in the original Fock space of linearized gravity over the de Sitter group. Then a finite inner product of these states is defined by dividing the original inner product by the infinite volume of the de Sitter group. The Hilbert space of de Sitter-invariant states thus obtained is hoped to serve as a starting point toward a meaningful perturbative quantum gravity (at the tree level) in de Sitter spacetime.