Elaan of Troyius established:
SULU: Captain, the Klingon ship is closing on an intercept course. Five hundred thousand kilometres. Deflector shields up.
So, we know it is greater than five hundred thousand kilometers. Prior to this scene, the Enterprise was able to image the Klingon warship at "extreme magnification", so, I assume it was greater than that distance.
The Ultimate Computer shows:
SPOCK: Captain, sensors report two contacts now. One on the port bow, one on the stern. Distance, two hundred thousand kilometres and closing.
I assume the ships were closing very fast and by the time Spock reported the sensor report, they moved well within the range limits.
In
The Changeling:
KIRK: Then we can't out run them. Good, Scotty. You're doing the right thing. (another hit) Source, Spock.
SPOCK: Unknown, Captain. Nothing within sensor range. (a third bolt approaching) Something now, Captain. Very small. Bearing one two three degrees, mark one eight. Range ninety thousand kilometres.
KIRK: That's our target, Mister Sulu. Prepare photon torpedo.
Nomad was detected at only ninety thousand kilometers, but it was very small and never imaged on the view screen.