COVID-19, caused by SARS-CoV-2, which is a positive-sense, single-stranded RNA enveloped virus, emerged in late 2019 and was declared a worldwide pandemic in early 2020 causing more than 600 million infections so far and more than 6 million deaths in the world. Although new vaccines have been implemented, the pandemic continues to impact world health dramatically. Membrane fusion, critical for the viral entry into the host cell, is one of the main targets for the development of novel antiviral therapies to combat COVID-19. The S2 subunit of the viral S protein, a class I membrane fusion protein, contains the fusion domain which is directly implicated in the fusion mechanism. The knowledge of the membrane fusion mechanism at the molecular level will undoubtedly result in the development of effective antiviral strategies. We have used all-atom molecular dynamics to analyse the binding of the SARS-CoV-2 fusion peptide to specific phospholipids in model membranes composed of only one phospholipid plus cholesterol in the presence of either Na+ or Ca2+. Our results show that the fusion peptide is capable of binding to the membrane, that its secondary structure does not change significantly upon binding, that it tends to preferentially bind electronegatively charged phospholipids, and that it does not bind cholesterol at all. Understanding the intricacies of the membrane fusion mechanism and the molecular interactions involved will lead us to the development of antiviral molecules that will allow a more efficient battle against these viruses.