We study a class of higher derivative, nonlocal gravity with a cosmological constant which admits homogeneous and isotropic nonsingular bouncing universes in the absence of matter. We show that, at the linearized level around the bounce, it is possible to constrain the nonlocal form factor such that there is only a scalar propagating degree of freedom, and no vector or tensor modes. The scalar mode can be made free from perturbative ghost instabilities and has oscillatory and bounded evolution across the bounce.