Air carrier Captain reported the L Bleed failure during cruise. After completing the QRH procedures and collaborating with Maintenance the flight crew returned to the departure airport.

Narrative

We had an MEL 36-XX-XXX. Leg 1 ZZZ- ZZZ1 was conducted normally with the MEL. Flight ZZZ1-ZZZ at cruise Altitude FL310 which is the limit for operating with the MEL we received an Amber Caution message L Bleed Fail. The Emergency Checklist triggered us to turn off L Bleed; APU Bleed; ManX Bleed along with Left Pack. There is a note in Emergency Checklist that says avoid and exit icing conditions due to single pack operation. We SATCOM dispatch and Maintenance Control which was approximately [a] 10 minute call and there reset was unsuccessful. The air synoptic page was showing the valve open with an Amber box and 9 PSI even though the valve was closed on the overhead panel. Dispatch said due to weather in ZZZ let's go back to ZZZ1. As we were getting setup for the air return the message cleared and we contacted dispatch immediately and elected to press on to ZZZ. Within 5 minutes the message came back and we contacted dispatch and this time we agreed to air return to ZZZ1 due to weather and our FOB was still good. We informed inflight as to the air return and made the customers aware of the situation. We landed in ZZZ1 normally and proceeded to the gate.