Nigel is selling candy bars at a rate of 4 bars every 3 days. To find out how many days it will take him to sell 36 bars, we can first determine how many sets of 4 bars are in 36 bars.
To calculate this:
\[ \text{Number of sets} = \frac{36 \text{ bars}}{4 \text{ bars per set}} = 9 \text{ sets} \]
Since each set takes 3 days to sell, the total time to sell 9 sets is:
\[ \text{Total days} = 9 \text{ sets} \times 3 \text{ days per set} = 27 \text{ days} \]
Thus, it will take Nigel a total of \( \boxed{27} \) days to sell his box of 36 candy bars.