A future multi-TeV muon collider would provide an important probe for Majorana neutrinos. A muon collider with a collision energy of $\sim$30 TeV would be sensitive to $\nu_e-\nu_\mu$ transition dipole moments of the order of $\sim 10^{-12}\mu_B$ and would be thus competitive with the latest astrophysical observation and laboratory experiments. Contrary to the latter, the muon collider would have the unique advantage of a direct and clean identification of lepton number and flavour violation. This would establish the Majorana nature of the neutrino and would provide crucial complementary information on the neutrino properties in the event of a (near) future observation at low-energy experiments. Additionally, a muon collider would improve by orders of magnitude the direct bounds on the Majorana neutrino mass matrix entries $m_{e\mu}$ and $m_{\mu\mu}$.