A nonlocal gravity model, which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator, is considered. The model is proven to have de Sitter solutions only if the function f satisfies a certain second-order linear differential equation. The de Sitter solutions corresponding to the simplest case, an exponential function f, are explored, without any restrictions on the parameters. If the value of the Hubble parameter is positive, the de Sitter solution is stable at late times, both for negative and for positive values of the cosmological constant. Also, the stability of the solutions with zero cosmological constant is discussed and sufficient conditions for it are established in this case. New de Sitter solutions are obtained which correspond to the model with dark matter and stability is proven in this situation for nonzero values of the cosmological constant.

20 pages, version to appear in Phys. Rev. D