B737 First Officer reported a fuel imbalance developed in cruise. Flight crew shut down an engine; rebalanced fuel and diverted to a landing.

Narrative

While cruising at FL 400; we noticed an imbalance in the wing tanks. At the time we were burning fuel from the center tank and the wing tanks should have both been full. Not long after; the IMBAL Alert illuminated. We began the QRH and started assessing our fuel plan. By this point our #1 Wing Tank was approximately 1;000 lbs less than the #2 Wing Tank. We noticed that we were going to be about 2000 pounds less than the fuel remaining at ZZZ waypoint ahead. Following the QRH it led us to the Engine Fuel Leak Checklist and continued to the One Engine Inoperative Checklist.After shutting the engine down; the fuel in the #1 Wing Tank no longer was decreasing and we were able to balance the fuel by burning only from the #2 Wing Tank. We [requested priority handling] and diverted to ZZZ. Upon landing in ZZZ we stopped on the runway and had CFR (Crash Fire Rescue) inspect the aircraft. After their assessment; they said that there was not a leak detected and they followed us to the gate. CFR evaluated the aircraft at the gate and deemed that no leaks were detected before leaving. Of note; we attempted to notify Dispatch via ACARS but did not complete the task due to time compression as we diverted to ZZZ.