Current sensing is performed in a non-volatile storage device for a selected non-volatile storage element with a negative threshold voltage. A control gate read voltage is applied to a selected word line of a non-volatile storage element, and source and p-well voltages are applied to a source and a p-well, respectively, associated with the non-volatile storage element. The source and p-well voltages exceed the control gate read voltage so that a positive control gate read voltage can be used. There is no need for a negative charge pump to apply a negative word line voltage even for sensing a negative threshold voltage. A programming condition of the non-volatile storage element is determined by sensing a voltage drop which is tied to a fixed current which flows in a NAND string of the non-volatile storage element.