Did you mean 2M Ohm? 2m Ohm is pretty small...
So you've boosted the gain by a factor of 20. To about 1000. That's a bit extreme in one stage.
Are you using an LMV358, or LM358? Not all Op Amps are that well behaved when running the inputs down to and below the negative rail.
It is also possible that it is going into oscillation after the input kicks the voltage a bit higher. Try putting a 100pF capacitor across R5 and see what happens.