High frequency electrometer
Fahgeddaboudit.

Do the math.  The capacitance to roll off at 10 MHz with a 100 M&Omega; impedance is only 160 fF.  There is basically no chance you can keep the parasitic capacitance that low with any off the shelf chip, and then there is whatever parasitic capacitance in the thing that is originating the signal, and in the connection between the thing and the chip intended to receive this signal.

The usual way to receive high impedance signals is to run them directly into the gate of a FET.  However, whether MOSFET or JFET, those will have substantially more gate capacitance than 160 fF.  The last resort is usually a vacuum tube, but even those have more grid capacitance than that.  I didn't look for low grid capacitance, but even a medium mu amplifier like a 12AU7-A has 1.5 pF grid to anode and 1.6 pF grid to cathode.  That's still 20x more than you can tolerate, even with a circuit that holds the plate voltage fixed.  Perhaps there are some specialty tubes that can do ultra-low grid capacitance, but 160 fF is a stretch.

Added:

You had originally asked for a very high impedance amplifier, but now you say it is permissible to short the signal.  That makes a huge difference.  Now you can measure the short circuit current instead of the open circuit voltage.  As others have said, you want a transimpedance amplifier, which means it takes current in and produces a proportional voltage out.

Your specs are still demanding because the signal is so tiny.  There will be lots of inherent sources of noise at that level.  You may need to actively cool the amplifier front end.

You said the impedance of this source is 100 M&Omega; and the open circuit voltage of the desired signal would be microvolts.  1 &micro;V / 100 M&Omega; = 1 fA.  That's tiny.

Rethink

You need to step back and look at the overall problem, and not pre-suppose a particular solution.  What are you really trying to measure.  Or perhaps pop up another level and explain what you are trying to control, prove, or whatever, that you think making this measurement is part of.