A nonlocal gravity model is considered which does not assume the existence of a new dimensional parameter in the action and includes a function f(□-1R), with □ the d'Alembertian operator. Using a reconstruction procedure for the local scalar-tensor formulation of this model, a function f is obtained for which the model exhibits power-law solutions with the Hubble parameter H = n/t, for any value of the constant n. For generic n - namely except for a few special values which are characterized and also specifically studied - the corresponding function f is a sum of exponential functions. Corresponding power-law solutions are found explicitly. Also the case is solved in all detail of a function f such that the model contains both de Sitter and power-law solutions. © 2013 IOP Publishing Ltd.