A calculation for the coherent scattering and conductance of magnons via atomic nanocontacts is presented. The model system is composed of two groups of semi-infinite magnetically ordered Heisenberg monatomic chains, joined together by the magnetic nanocontact, and the system is supported on a non-magnetic substrate and considered otherwise free from magnetic interactions. The coherent transmission and reflection coefficients are derived as elements of a Landauer-type scattering matrix. Transmission and reflection scattering cross sections are calculated specifically for three distinct symmetric and asymmetric geometric configurations of the nanocontact. Three cases of local magnetic exchange on the nanocontact domain are analysed for each configuration to investigate the influence of softening and hardening of the magnetic boundary conditions. In analogy with coherent electronic transport, we calculate the magnon coherent transport. The numerical results show the interference effects between the incident scattered magnons and the localized spin states on the nanocontact, with characteristic Fano resonances. The numerical results yield an understanding of the relationship between the coherent magnon conductance and the architecture of the embedded magnetic nanocontact.