B737-800 Captain reported a fuel imbalance indication during climb out and the need to return to the departure airport.

Narrative

I was the pilot flying on Aircraft X; ZZZ-ZZZZ. While completing the After Takeoff Checklist; my First Officer (FO) called my attention to a FUEL IMBAL indication on fuel quantity indicator. The left main tank was indicating about 1;000 pounds less than the right main tank and we were also burning center tank fuel. Having taken off with full mains and roughly 14;000 pounds in the center tank; we knew we should not be seeing an imbalance indication. I was hand flying at the time and called for the FUEL IMBALANCE CHECKLIST. As she was pulling up the checklist on the QRH; I tried engaging the Autopilot on CMD A so I could transfer control and run the checklist. It didn't engage so I continued to hand fly and had her run the checklist. The fuel level rate decrease in the left main was alarming which led us to the FUEL LEAK ENGINE CHECKLIST. I tried re-engaging the Autopilot on CMD B which successfully engaged this time. I informed ATC of our situation and we may need to return to ZZZ. As per the checklist; we turned off the center tank fuel pumps and the fuel level rate decrease immediately slowed. I called the flight attendants to see if they noticed fuel spray out of the left engine/wing/pylon area. They called back and said no. The checklist led us to turning the center tank pumps back on. The engines seemed to be mostly burning out of the center tank however we noticed the left main began decreasing again; however this time at a much lower rate. I transferred aircraft control to the FO while I attempted to call Dispatch on the local ZZZ frequency. They didn't respond so I tried to call Maintenance to see if they was anything we could do. They offered no suggestions. The fuel level was still decreasing out of the left main. It was obvious we could not continue on to ZZZZ nor could we afford to burn fuel down to below max landing weight as the fuel imbalance was out of limits and worsening. I advised ATC and informed them of our need to return. I informed the flight attendants; passengers; and completed the non routine landing checklist; reprogrammed the FMS; and advised Dispatch via ACARS; computed landing performance. I took control of the airplane and we returned for an overweight landing. After clearing the runway; the Airport Rescue and Firefighting (ARFF) vehicles inspected the aircraft and saw no visible signs of a fuel leak. We taxied to the gate uneventfully.