The theory of liquid fuel jet instabilities has been developed under several assumptions, which include the assumption that the jets breakup processes are isothermal. However, liquid fuels are normally injected into an elevated combustion-chamber temperature to maintain a desirable homogeneous combustible mixture - liquid vapour and air. Therefore, a new linear theory model is developed for the instability and breakup of non-isothermal liquid jets, with consideration of a spatially variation of surface tension along the liquid-gas interface. The spatial variation of surface tension is obtained through temperature-dependent surface tension and transient heat-transfer from the combusting gases to the liquid jet. The classical interface hydrodynamic breakup theory and solution of heat-transfer through semi-infinite medium are coupled through the surface tension gradient. The analytical model accounts for the non-isothermal effects on jet breakup. The model is shown to maintain plausibility of investigating liquid fuel breakup mechanisms under the combustion-chamber operating conditions.