The presence of live $^{60}$Fe nuclei (lifetime of 3.8~Myr) in cosmic rays detected by the ACE/CRIS instrument suggests a nearby nucleosynthesis source. $^{60}$Fe is primarily produced in core-collapse supernovae, and we aim to clarify whether the detected $^{60}$Fe nuclei can be associated with a particular local supernova. We consider 25 OB associations and sub-groups located within 1 kpc of the solar system based on recent $Gaia$ census. A model is developed that combines stellar population synthesis within these OB associations, cosmic-ray acceleration within associated superbubbles, and cosmic-ray transport to the solar system. The most critical model parameter impacting $^{60}$Fe cosmic-ray production is the explodability criterion, which determines if a massive star ends its life as a supernova. Our study points to the Sco-Cen OB association as the most probable origin of the observed $^{60}$Fe nuclei, particularly suggesting they were accelerated in the Sco-Cen superbubble by a young supernova aged $\leq500$ kyr with a progenitor mass of approximately $13-20~M_\odot$. A less likely source is the supernova at the origin of the Geminga pulsar 342 kyr ago, if the progenitor originated in the Orion OB1 association. The contribution of local OB associations to the cosmic-ray density of stable $^{56}$Fe is estimated to be around 20%, with some sensitivity to cosmic ray acceleration efficiency and diffusion coefficient. These findings shed light on the origins of cosmic-ray nuclei, connecting them to nucleosynthesis events within our local cosmic neighborhood.