An obvious potential candidate for the job? Former Patriots coach Bill Belichick is out there, lurking in the weeks, waiting for another opportunity to coach. But is the Saints job right for Belichick? They’ll kick the tires. Why wouldn’t they?

It’s not uncommon for out-of-work coaches to pursue television commentary, but Mr. Belichick was renowned for being boring and cryptic. His mid-week and post-game press sessions were marked by short non-answers with a large dose of grumpiness.