10M Ω implies tiny Amperes. Very often neighboring devices (example, IC's, op amps) have 1MΩ input impedance. As well as common DMM's. Your RC network is likely to be 'loaded' by such devices, throwing off expected volt levels. The tiny Ampere level may be insufficient to drive another device.
In real life you can discover a range of your 10M potentiometer where you can dial the resistance that provides the needed RC time constant. However I think the 1M potentiometer is a better choice. Does your project run on batteries? Then you might prefer to look for ways to minimize power usage. It depends on what trade-offs you decide to make.