Summary: The recently discovered centre-mode instability of rectilinear viscoelastic shear flow [P. Garg et al., “Viscoelastic pipe flow is linearly unstable”, Phys. Rev. Lett. 121, No. 2, Article ID 024502, 6 p. (2018; doi:10.1103/PhysRevLett.121.024502)] has offered an explanation for the origin of elasto-inertial turbulence that occurs at lower Weissenberg numbers (\(Wi\)). In support of this, we show using weakly nonlinear analysis that the subcriticality found in [the second author et al., “Exact traveling wave solutions in viscoelastic channel flow”, Phys. Rev. Lett. 125, No. 15, Article ID 154501, 5 p. (2020; doi:10.1103/PhysRevLett.125.154501)] is generic across the neutral curve with the instability becoming supercritical only at low Reynolds numbers (\(Re\)) and high \(Wi\). We demonstrate that the instability can be viewed as purely elastic in origin, even for \(Re=O(10^3)\), rather than ‘elasto-inertial’, as the underlying shear does not feed the kinetic energy of the instability. It is also found that the introduction of a realistic maximum polymer extension length, \(L_{\max}\), in the FENE-P model moves the neutral curve closer to the inertialess \(Re=0\) limit at a fixed ratio of solvent-to-solution viscosities, \(\beta\). At \(Re=0\) and in the dilute limit (\(\beta \rightarrow 1\)) with \(L_{\max} = O(100)\), the linear instability can be brought down to more physically relevant \(Wi \gtrsim 110\) at \(\beta = 0.98\), compared with the threshold \(Wi=O(10^3)\) at \(\beta = 0.994\) reported recently by M. Khalid et al. [“Continuous pathway between the elasto-inertial and elastic turbulent states in viscoelastic channel flow”, Phys. Rev. Lett. 127, No. 13, Article ID 134502, 6 p. (2021; doi:10.1103/PhysRevLett.127.134502)] for an Oldroyd-B fluid. Again, the instability is subcritical, implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable – i.e. unstable to finite-amplitude disturbances – for even lower \(Wi\).