Step 1: Understanding the geographical context.
- Agra is located on the banks of the Yamuna river.
- Patna is located on the banks of the Ganges river.
- Kolkata is located on the banks of the Hooghly river.
- Bhopal, however, is not located on any major river but rather has a network of lakes, with no direct proximity to a large river like the others.
Step 2: Conclusion.
Among the cities listed, Bhopal is the only one not situated on the banks of a major river.
\[
\boxed{\text{Thus, the correct answer is Bhopal.}}
\]