The scattering model of a single high-energy electron interacting with a Gaussian laser pulse is constructed on the basis of Lagrange’s equation to explore the relationship between high-energy electron radiation and the transverse initial position of electrons in tightly focused intense laser pulse line polarization. And data on electron motion and electron radiation were obtained with the help of MATLAB. After the interaction between the high-energy still electrons and strong tightly focused linearly polarized laser, they first oscillate in xOz plane, and then make a linear motion in the positive direction of the z-axis, with its lateral velocity gradually converting to the longitudinal velocity. Observing the interesting motion trajectory, spectral modulation structure, and spatial radiation distribution, it is found that the transverse initial position of the electron shows great impacts on electron movement, radiation intensity, and radiation distribution. As the transverse initial position of the electron increases positively, the direction of electron motion is rapidly shifted by the z-axis, and the transverse and vertical velocity loses its symmetry as well; there exist maximum values of the maximum spectral amplitude and the maximum radiation intensity with the increase of the transverse initial position, of which the radiation direction keeps its azimuth angle unchanged and the polar angle expanding. It is demonstrated that the center of laser pulse is not optimal for obtaining the brightest and hardest source of electromagnetic radiation. The hardest radiation pulse is generated when the initial position is set to 0.085 , with the maximum radiation energy at 1.34×10^8 , the polar angle is about 17 °, and the azimuth angle is 0 °