B757 Captain reported both left fuel boost pumps failed during cruise. The flight crew requested priority handling and descended to follow the QRH procedures. The Flight crew requested direct to destination airport and made a precautionary landing.

Narrative

We were enroute from ZZZ to ZZZ1. Cruising at FL410 a little over 2 hours flying time. The FWD and AFT left pressure lights illuminated on the overhead panel and begin to flicker; slowly at first; then becoming quicker to an almost steady flash. My FO (First Officer) pulled out the QRH (Quick Reference Handbook) to reference FWD and AFT fuel pump; fuel configuration; and fuel system pressure. To prevent a flame out; I simultaneously opened the cross feed valves. The pressure lights extinguished. We closed the cross feed valves to see if the issue had resolved itself but the pressure lights returned in 8 seconds. At which time we reopened the cross feed valves. While we read through the QRH (Quick Reference Handbook); we recognized another issue developing rather quickly; a fuel imbalance. We would be outside of our 1;950 pounds structural imbalance limitation in a matter of minutes in the current configuration. We discussed and decided to descend to FL250 to allow the left engine to suction/gravity feed. Upon reaching FL250; I closed the cross feed valves. The low pressure lights came on immediately and began to flicker quickly like before; and the fuel flow remained solid around 3.3 on both sides - the engine was presenting positive indications of being gravity fed.In this time; we had reached a fuel imbalance of 1;300 pounds between the left and right tanks. A brief observation proved that both motors were now burning fuel from the respective tanks equally. The Left FWD/AFT fuel pump EICAS (Engine Indicating and Crew Alerting System) message would occasionally present it self. We discussed the possibility of having FOD'ed out the left fuel pumps from debris in the left tank versus potentially having received contaminated fuel which could endanger the right engine. From the information in front of us; we settled on the first school of thought. We requested priority handling and proceeded direct to ZZZ1. The situation remained the same to an uneventful landing on XXC. I suspect possible debris in the left fuel tank became lodged in the FWD and AFT fuel pump impellers; creating a low fuel pressure state. When the cross feed valves were open; there was enough pressure being routed from the right side to prevent the low pressure light from activating and preventing a flame out. Below FL250; we were able to gravity feed the left engine.