We study the coupled spatio-temporal variations of the electron density and the electric field (electron plasma oscillations) in high-electron mobility transistors using the developed device model. The excitation of electron plasma oscillations in the terahertz range of frequencies might lead to the emission of terahertz radiation. In the framework of the model developed, we calculate the resonant plasma frequencies and find the conditions for the plasma oscillations self-excitation (plasma instability) We show that the transit-time effect in the high-electric field region near the drain edge of the channel of high-electron mobility transistors can cause the self-excitation of the plasma oscillations. It is shown that the self-excitation of plasma oscillations is possible when the ratio of the electron velocity in the high field region, ud, and the gate length, Lg, i.e., the inverse transit time are sufficiently large in comparison with the electron collision frequency in the gated channel, ν. The transit-time mechanism of plasma instability under consideration can superimpose on the Dyakonov-Shur mechanism predicted previously strongly affecting the conditions of the instability and, hence, terahertz emission. The instability mechanism under consideration might shed light on the origin of terahertz emission from high electron mobility transistors observed in recent experiments.