My charger is connected to a power outlet, the power source shows as power adapter, but shows that battery is not charging.

Is your battery charged between 90% and 100%? If so, that may be normal behavior. Apple's battery management will sometimes allow the battery to drain from 100% to 90% and then charge back up to 100%, this preserves the battery better than just leaving it charged to 100% all the time.

If, while connected to the charger, your battery drops below 90% then that is not normal and your charger or its cable may be broken.