We discuss a left-right (L-R) symmetric model with the double seesaw mechanism at the TeV scale generating Majorana masses for the active left-handed (LH) flavour neutrinos ν αL and the heavy right-handed (RH) neutrinos N βR , α, β = e, µ, τ , which in turn mediate lepton number violating processes, including neutrinoless double beta decay. The Higgs sector is composed of two Higgs doublets H L , H R and a bi-doublet Φ. The fermion sector has the usual for the L-R symmetric models quarks and leptons, along with three SU (2) singlet fermion S γL . The choice of bare Majorana mass term for these sterile fermions induces large Majorana masses for the heavy RH neutrinos leading to two sets of heavy Majorana particles N j and S k , j, k = 1, 2, 3, with masses m N j m S k . Working with a specific version of the model in which the ν αL − N βR and the N βR − S γL Dirac mass terms are diagonal, and assuming that m N j ∼ (1 − 1000) GeV and max(m S k ) ∼ (1 − 10) TeV, m N j m S k , we study in detail the new "non-standard" contributions to the 0νββ decay amplitude and half-life arising due to the exchange of virtual N j and S k . We find that in both cases of NO and IO light neutrino mass spectra, these contributions are strongly enhanced and are dominant at relatively small values of the lightest neutrino mass m 1(3) ∼ (10 −4 − 10 −2 ) eV over the light Majorana neutrino exchange contribution. In large part of the parameter space, the predictions of the model for the 0νββ decay generalised effective Majorana mass and half-life are within the sensitivity range of the planned next generation of neutrinoless double beta decay experiments LEGEND-200 (LEGEND-1000), nEXO, KamlAND-Zen-II, CUPID, NEXT-HD.