Air carrier Captain reported engine #1 overheat light illuminated during climb. The flight crew completed the QRH procedures and were provided priority handling to departure airport where a safe landing was completed.

Narrative

On push back from the gate; on a very hot day in ZZZ; we started both engines. I instructed the FO (First Officer) to place the bleeds in the Hot Weather configuration; with the APU on. As he was moving the ENG 1 Bleed air switch off the ENG 1 Overheat light illuminated and then went off. We put the bleed back on and tried to get a recall but nothing came up. All engine instrumentation read out normal; so we thought it was a fluke. Everything was normal on taxi out and take off. With the engines at climb power and climbing through roughly 5000 ft; the ENG 1 overheat light illuminated along with a master caution. My FO continued flying; I ran the QRC. We leveled off at 8000 ft; [requested priority handling] and got vectors back to ZZZ. The QRC had me turn the A/T (autothrottles) off and retard the number 1 Eng to idle. After a few seconds the overheat light extinguished. All engine instruments were indicating normal. I continued to the QRH; once finished we returned to ZZZ and landed with ENG 1 at idle. Fire and Rescue met us on the runway and gave us the all clear. We then taxied back to the gate. Cause: Possibly a fault somewhere in the overheat detection system; exasperated by the extreme heat. Suggestion: The only change I would have made was to call Maintenance after the initial occurrence; though I am not sure they would have been able to do anything or get the aircraft to recreate the fault.