The linear stability of Burgers and Lamb–Oseen vortices is addressed when the vortex of circulation Γ and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting non-axisymmetric vortex is analysed in the limit of large Reynolds number RΓ=Γ/v and small strain s[Lt ]Γ/δ2 by considering the approximations obtained by Moffatt et al. (1994) and Jiménez et al. (1996) for each case respectively. For both vortices, the TWMS instability (Tsai & Widnall 1976; Moore & Saffman 1975) is shown to be active, i.e. stationary helical Kelvin waves of azimuthal wavenumbers m=1 and m=−1 resonate and are amplified by the external strain in the neighbourhood of critical axial wavenumbers which are computed. The additional effects of diffusion for the Lamb–Oseen vortex and stretching for the Burgers vortex are proved to limit in time the resonance. The transient growth of the helical waves is analysed in detail for the distinguished scaling s∼Γ/ (δ2R1/2Γ). An amplitude equation describing the resonance is obtained and the maximum gain of the wave amplitudes is calculated. The effect of the vorticity profile on the instability characteristic as well as of a time-varying stretching rate are analysed. In particular the stretching rate maximizing the instability is calculated. The results are also discussed in the light of recent observations in experiments and numerical simulations. It is argued that the Kelvin waves resonance mechanism could explain various dynamical behaviours of vortex filaments in turbulence.