We used a sample of gamma-ray bursts (GRBs) detected by Fermi and Swift to reanalyze the correlation discovered by Amati et al. (2002A&A...390...81A) between E_pi_, the peak energy of the prompt GRB emission, and E_iso_, the energy released by the GRB assuming isotropic emission. This correlation has been disputed by various authors, and our aim is to assess whether it is an intrinsic GRB property or the consequence of selection effects. We constructed a sample of Fermi GRBs with homogeneous selection criteria, and we studied their distribution in the E_pi_-E_iso_ plane. Our sample is made of 43 GRBs with a redshift and 243 GRBs without a redshift. We show that GRBs with a redshift follow a broad E_pi_-E_iso_ relation, while GRBs without a redshift show several outliers. We use these samples to discuss the impact of selection effects associated with GRB detection and with redshift measurement.