Arvo,
I am not sure from your description how exactly your configuration looks like - is it simply a follower with 100M resistor in series with the positive input terminal? What do you mean by drift - it is not a temperature drift you have in mind, is it? Do you mean a voltage drop due to the input bias current passing through 100M input resistor?
The maximum input bias current of OPA277 is 4nA (see below) thus you should only see 4mV with 1M input resistor, or 400mV with 100M, unless the your ambient temperature is above 85C or the circuit configuration is unstable - check the output with a scope for any sign of oscillation. Do you factor in the OPA277 finite input impedance? Do you use decoupling capacitors (0.1uF) on the supply pins?
If you need more help, please send your complete schematic including all the values of external components and supply voltages so we may evaluate it not only from a DC biasing but also from a stability point of view.
