In recent sawtooth experiments at JET fast electron bremsstrahlung was found to be significantly enhanced in low-density JET plasmas with grassy sawteeth [1]. Similar observations were found in T-10 where the appearance of beams of suprathermal electrons accompanied the sawteeth [2]. Estimates for grassy sawteeth JET discharges show that the on-axis inductive electric field is close to the critical electric field corresponding to the minimum of the friction force on a relativistic electron, while the electric field induced by sawteeth in the reconnection region could approach the Dreicer electric field. Under such conditions, suprathermal electrons are generated at each sawtooth crash while a near-threshold inductive electric field prevents a rapid deceleration of these electrons. In this paper, a corrected relativistic collision operator is used to derive a Fokker- Planck equation for the distribution function of relativistic suprathermal electrons in a weakly relativistic plasma, Te /mec2 ~ 10-2 << 1, which is then solved by a procedure similar to that employed in [3] and [4]. Special emphasis is placed on a near-critical electric field case typical for plasmas with grassy sawteeth on JET. It is found that the main result concerning the runaway rate is still valid. In addition [4], new near-critical electric field regions are considered. In the weakly relativistic region a numerical solution enables matching to the high momentum analytical asymptote. The form of the electron velocity distribution function up to the relativistic region, where suprathermal electrons are present, is thus accurately described and can be used for investigations of fast electron bremsstrahlung from suprathermal electrons in hot tokamak plasmas.