Voltage drop at input of opamp
Edited question for non-inverting case, removed circuit + analysis of negative op-amp amplifier.

Yes the problem is most likely the 10MOhm resistor.  This happens because while the op-amp assumption is that $$ I_+ = I_- = 0 $$ this is not really the case, and there is a small input current.  

You should decrease the size of both voltage divider resistors while keeping their ratio the same.  This will give the required attenuation while also allowing the op-amp terminals to sink some current.  

You can look at the datasheet to see how much current the op-amp inverting/non-inverting terminals actually draw.  You want this current to make only a small difference compared to the voltage you are amplifying.

I will say that trying to amplify a 500uV signal like this is going to create a lot of noise, and the voltage divider will draw a lot of power.  A better way is probably to control the gain with the feedback resistor ratio rather than pre-attenuate it like this, especially for a 1V signal.