Observations show that massive star formation quenches first at the centres of galaxies. To understand quenching mechanisms, we investigate the thermal and non-thermal energy balance in the central kpc of NGC 1097—a prototypical galaxy undergoing quenching—and present a systematic study of the nuclear star formation efficiency and its dependencies. This region is dominated by the non-thermal pressure from the magnetic field, cosmic rays and turbulence. A comparison of the mass-to-magnetic flux ratio of the molecular clouds shows that most of them are magnetically critical or supported against the gravitational collapse needed to form the cores of massive stars. Moreover, the star formation efficiency of the clouds drops with the magnetic field strength. Such an anti-correlation holds with neither the turbulent nor the thermal pressure. Hence, a progressive build up of the magnetic field results in high-mass stars forming inefficiently, and this may be the cause of the low-mass stellar population in the bulges of galaxies.