Hi,
Just to add a couple notes here.
First, a non inverting amplifier stage is 1+Rf/Rin so with 1M and 10k we have 1+100 which is a gain of 101.
With 1mv input offset that means 0.101 volts output. With 2mv input offset that means 0.202 volts output. I've never seen one of these op amps with an input offset greater than 2mv at 25 degrees C ambient, and that is probably because at other temperatures it could go even higher. If the max was 5mv though it would still only be 0.505 volts output, and all these are plus and minus.
To get +2v on the output we'd have to see about 20mv input offset, which would mean a bad op amp chip. If there are other components on the input however, we could see effects from those parts also. Schottky diodes are notorious for higher leakage and can look like a 20k resistor with a reverse voltage. Compare that to the 500M reverse resistance of a 1N4148 diode.
So how much leakage does it take to force 20mv extra on the input of the input impedance is 10k?
It takes a whopping 2ua to get an extra 20mv across a 10k resistor! Ha!
So what input voltage could cause that with a 20k resistor? Well, about 40mv.
So with just 40mv and some extra leakage we could see an extra 20mv input.
This analysis is approximate to just get some idea what it takes to mess up a seemingly good op amp circuit.
I've seen worse. I've actually seen an inexplicable near zero output offset when it could have been more like 10mv.
I've never seen a higher than expected output offset without a good reason for it however.