In this work, the graphene wormhole in two cases, i) free case and ii) in the presence of an Aharonov–Bohm (AB)-like magnetic field are studied. To achieve the objective, the massless Dirac equation is solved in the background of a catenoid metric. As this system does not support exact solutions due to the considered background, the scattering-state solutions are obtained by means of the Born approximation. In both cases, the Landauer formalism is used to obtain approximate expressions for the conductance. For the free case, a minimum conductance, which is independent of system parameters is found. For the case with an AB-like magnetic field, the results show that the conductance can be modulated by means of the parameter related to magnetic flux. In addition, a special case where a null magnetic flux generates an oscillatory conductance is analyzed.