The short answer is no. That assumes you’re using a standard style MUX like the CD74HC4067.
The longer answer is that the mega328 does have an ADC complete interrupt, so you could fake it by using the ADC conversion interrupt to switch to the next MUX channel and start a conversion. Between interrupts you still could (theoretically) do other things. You probably want to read section 23 from the mega328’s datasheet before you go too much farther. Alternatively you could use a timer interrupt, to periodically loop through all the MUX channels. It all depends on how much lag you’re willing to accept.