We investigate the propagation of magneto-thermoelastic interaction in an isotropic homogeneous perfectly conducting two-dimensional semi-infinite medium using a new theory of two-temperature generalized thermoelasticity with memory-dependent derivative. The bounding surface of the medium is taken as stress free and subjected to a time dependent thermal shock. The combined Laplace–Fourier transform is applied to find the solutions in transform domain. Numerical results of the field variables are presented graphically to discuss the effect of time, space variable and the time-delay for some specific cases. The graphical representation shows that time-delay has significant effect on the distribution on the physical variables. © 2018, © 2018 Taylor & Francis Group, LLC.