A search for Majorana neutrinos in same-sign WW scattering events is presented. The analysis uses $\sqrt{s}= 13$ TeV proton–proton collision data with an integrated luminosity of 140 fb$^{-1}$ recorded during 2015–2018 by the ATLAS detector at the Large Hadron Collider. The analysis targets final states including exactly two same-sign muons and at least two hadronic jets well separated in rapidity. The modelling of the main backgrounds, from Standard Model same-sign WW scattering and WZ production, is constrained with data in dedicated signal-depleted control regions. The distribution of the transverse momentum of the second-hardest muon is used to search for signals originating from a heavy Majorana neutrino with a mass between 50 GeV and 20 TeV. No significant excess is observed over the background expectation. The results are interpreted in a benchmark scenario of the Phenomenological Type-I Seesaw model. In addition, the sensitivity to the Weinberg operator is investigated. Upper limits at the 95% confidence level are placed on the squared muon-neutrino–heavy-neutrino mass-mixing matrix element $\vert V_{\mu N} \vert ^{2}$ as a function of the heavy Majorana neutrino’s mass $m_N$, and on the effective $\mu \mu $ Majorana neutrino mass $|m_{\mu \mu }|$.