Molecular dynamics simulations in the NPT ensemble have been carried out to investigate the effect of two room temperature ionic liquids (RTILs), on stacks of phospholipid bilayers in water. We consider RTIL compounds consisting of chloride ([bmim][Cl]) and hexafluorophosphate ([bmim][PF₆]) salts of the 1-buthyl-3-methylimidazolium ([bmim]⁺) cation, while the phospholipid bilayer is made of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC). Our investigations focus on structural and dynamical properties of phospholipid and water molecules that could be probed by inelastic and quasi-elastic neutron scattering measurements. The results confirm the fast incorporation of [bmim]⁺ into the lipid phase already observed in previous simulations, driven by the Coulomb attraction of the cation for the most electronegative oxygens in the POPC head group and by sizeable dispersion forces binding the neutral hydrocarbon tails of [bmim]⁺ and of POPC. The [bmim]⁺ absorption into the bilayer favours the penetration of water into POPC, causes a slight but systematic thinning of the bilayer, and further stabilises hydrogen bonds at the lipid/water interface that already in pure samples (no RTIL) display a lifetime much longer than in bulk water. On the other hand, the effect of RTILs on the diffusion constant of POPC (D_POPC) does not reveal a clearly identifiable trend, since D_POPC increases upon addition of [bmim][Cl] and decreases in the [bmim][PF₆] case. Moreover, because of screening, the electrostatic signature of each bilayer is only moderately affected by the addition of RTIL ions in solution. The analysis of long wavelength fluctuations of the bilayers shows that RTIL sorption causes a general decrease of the lipid/water interfacial tension and bending rigidity, pointing to the destabilizing effect of RTILs on lipid bilayers.