The first measurement of the muon anomalous magnetic moment by the Fermilab g − 2 experiment has confirmed the previous intriguing result of the BNL experiment. Their combination brings to 4.2σ the discrepancy with the currently accepted prediction of the Standard Model. The dominant theory uncertainty is related to the leading order hadronic vacuum polarization contribution (LO- HVP), determined by a data-driven dispersive approach, using the hadron production cross section in e+e− annihilation. In contrast, a recent ab initio calculation of the LO-HVP contribution, based on Lattice QCD, weakens the discrepancy with the measurement, in some tension with the data- driven estimate. In this scenario, the novel approach proposed by the MUonE project aims at a third completely independent and competitive determination of the LO-HVP contribution, achievable with an alternative method based on the measurement of the effective electromagnetic coupling in the space-like region at low momentum transfer. We will discuss the possibility of performing this measurement at CERN by a very precise determination of the shape of the differential muon- electron elastic cross section, exploiting the scattering of 160 GeV muons on atomic electrons of a low-Z target. The project status will be presented, in view of the test run on a reduced detector expected to start at the end of 2021.