B737 Captain reported fuel imbalance due to possible fuel leak resulting in a diversion.

Narrative

At cruise I noticed a fuel imbalance; with the right tank being 1500 lbs lower than the left. Had the amber alert with no fuel in the center tank. Turned both right side pumps off and opened the cross feed. Then I noticed the fuel over destination being lower than normal; 7.2 was on the flight plan and it was showing 6.0. CI was planed at 120 and was not changed. The fuel over destination kept going down at approximately 500 lbs every 10 min. Also the right side tank kept losing gas with the pumps shut off; it did slow down but never stopped. Ran the FUEL LEAK-ENGINE QRH then talked to Dispatch on ARINC and advised about the idea of diverting. We both agreed ZZZ1 was a good choice. As we were getting farther away from land at this point; when the fuel over destination reached 5.7 and still falling we decided to divert to ZZZ1. Advised ATC; the flight attendants were briefed as well as the passengers on our situation and diversion. On downwind the tanks were in balance. At the marker the Left side now showed 1500 lbs low. No action was taken and we landed safely. Emergency vehicles inspected our aircraft and found no signs of leakage; so we taxied to the gate with vehicles following.