Step 1: Recall Nashik's location.
Nashik (also spelled Nasik) is a major city in Maharashtra, India, and holds great religious significance as one of the four Kumbh Mela sites.
Step 2: Identify the river.
The Godavari river, often called the "Dakshin Ganga" or Ganges of the South, flows through Nashik and is the second longest river in India.
Step 3: Eliminate wrong options.
Mahanadi flows through Chhattisgarh and Odisha, Tapi flows through Surat, and Krishna does not pass through Nashik.
\[ \boxed{\text{Godavari}} \]