Let G be a simple graph of order n. The resolvent Estrada index of G is defined as EEr = nΣi=1 (1- λi/(n-1)^-1, where λ1;λ2;…;λn are the eigenvalues of G. Formulas for computing EEr of the cycle Cn and the path Pn are derived. The precision of these approximations are shown to be excellent. We also examine the difference and relations between the Estrada index and the resolvent Estrada index of Cn and Pn.