You mean a pot in series on the +12 V rail as a rheostat?
I might be thinking about this all wrong, but: The LM358 needs 3 V minimum, so with a 12 V supply the maximum voltage drop across the pot would be 9 V. Each LM358 draws approximately 1 mA, so the circuit probably draws roughly 5 mA. That means the maximum value of R is 9 V / 5 mA = 1.8k. Using a 1k pot would get you a range from 7 to 12 V. A 500R pot would only get you 9.5 to 12 V. A 5k pot would take you from 12 V to down below that 3 V minimum.
Pots typically are rated for 1/8 W. (Check the datasheet.) With a 1k pot that corresponds to 11 mA maximum (at any wiper setting), so 5 mA should be safe. For a 5k pot the maximum current would be 5 mA, so if you really were drawing 1 mA per LM358 you’d be on the edge of burning out the pot.
So I think a 1k pot could work. Smaller wouldn’t give you much voltage range and larger might put the pot at risk, unless you find a higher power rated pot.
(If you use a 1k pot and it goes up in smoke, I will delete this post and deny everything.)