The invention relates to a linear constant current light-emitting diode (LED) driving circuit and an LED lamp. The LED lamp comprises an alternating current input power source, a bridge rectifier, the linear constant current LED driving circuit and an LED lamp body, wherein the linear constant current LED driving circuit comprises an output sampling module, a feedback control module, a drive control module, and a power control module; the output sampling module is used for collecting output current information of the LED lamp body; the feedback control module is used for generating and outputting integral voltage according to the output current information, wherein the integral voltage is kept constant in a power frequency period; the drive control module is used for generating and outputting a drive control signal according to the integral voltage; and the power control module is used for driving and controlling the LED lamp body in a constant-current mode according to the drive control signal and bearing voltage drop in a circuit. According to the linear constant current LED driving circuit and the LED lamp, constancy of average output current and average output power of the LED lamp body can be achieved.