variation in source votlage is allowed in a constant current circuit?
Here's the circuit linked in the question: -



It uses an LT6231 op-amp and, the front page of the data sheet tell us this (inside the red box): -




  I am seeing a variation in the load current as I change the voltage V2
  to and fro from 10V to 20V


You are exceeding the supply voltage of the device. Don't do this without expecting problems. Generally though, even if you weren't exceeding the supply voltage maximum, power supply rejection on all op-amps isn't infinite.