With the Standard Model being unable to describe the observed baryon asymmetry or the dark matter density in the universe, many models of New Physics introduce heavy neutrino species as a possible explanation for these effects. Dirac or Majorana neutrinos with masses above the electroweak scale could be produced at future linear $e^+e^-$ colliders, such as the Compact LInear Collider (CLIC) or the International Linear Collider (ILC). We studied the possibility of observing production and decays of the heavy neutrinos in the $qql$ final state at ILC running at 500 GeV and 1 TeV and CLIC running at 3 TeV. The analysis is based on the WHIZARD event generation and fast simulation of the detector response with DELPHES. Dirac and Majorana neutrinos with masses from 200 GeV to 3.2 TeV are considered. Estimated limits on the production cross sections and on the neutrino-lepton coupling are compared with the current limits coming from the LHC running at 13 TeV, as well as the expected future limits from hadron colliders. The obtained results are stricter than any other limit estimates published so far.