The linear stability of the Burgers and Lamb-Oseen vortices is addressed when the vortex of circulation C and radius r 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 Re = C / \nu and small strain s << C / r^2 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 neighborhood 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 details for the critical scaling s = C /(r ^2 Re^(1/2)) s_0 with s_0 =O(1). 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 discussed. The results leads to explicit sufficient conditions for the instability of arbitrarily stretched Gaussian vortices in a perpendicular strain field. It is also argued that this instability could explain various dynamical behaviors of vortex filaments in turbulence.