Why is my car not turning over

I have a 89 mustang with a 5.0. Motor, the problem I am having, is I can push start the car and it runs fine, but as soon as I shut it down it will not start up, I can hear the starter solenoid, the horn and lights work, when I turn the key on the fuel pump engages, I engage the clutch and all that I get is the sound of the starter solenoid. The starter will not engage none what so ever. Also there is a lot of heat under the engine bay