McCown, who has played well in place of the injured Jay Cutler, said that he would continue to play as if it were his last game in the NFL.
"One of the things we say all the time is, 'Don't let familiarity rob you of your joy,'" McCown told the club's official website. "So I think just not getting comfortable or taking any of these games for granted for me is the key.
"I appreciate being out there and if you appreciate being out there, you're going to put your best effort forward because you don't know when it's your last time.
"That's my mindset going into these games is I never want to get too comfortable. The minute you say you've got it, you'll be on your way out of here. It's definitely with an appreciative attitude that I take the field and enjoy playing."
McCown has also played for Carolina Panthers, Detroit Lions and Arizona Cardinals.