Diagnosis
The water inlet valve is stuck open. This solenoid-controlled valve lets hot and cold water in. If one of the little solenoids fails or if debris from your water lines holds the internal diaphragm open, water will slowly seep through even without electrical power.
How to Fix It
Turn off both hot and cold water supply lines at the wall. Unplug washer. The inlet valve is at the back, top-left corner, where the hoses connect. Disconnect the hoses and the wire harness. Remove the two screws holding it to the cabinet. The replacement part number is typically WPW10749472. Install the new one, reconnect hoses and wires.
If That Didn't Work
If the slow fill continues, the problem could be back-pressure from the drain system. Check that the drain hose isn't shoved too far down the standpipe—it can create a siphon. It should only be inserted 6-8 inches, and there should be an air gap.
Heads Up
Before installing the new valve, take a second to flush your supply hoses into a bucket to clear out any sediment that might wreck the new valve. Installing cheap inlet hose filters is a good idea.