A source said Bates was tempted to be reunited with Bears quarterback Jay Cutler, but the long-term security with Carroll was a major factor in his decision to join the Seahawks rather than the Bears. Bates was Cutler's offensive coordinator with the Broncos in 2008.
The Bears' options for the position now include former St. Louis Rams head coach Mike Martz, University of Minnesota offensive coordinator Jedd Fisch and Atlanta Falcons quarterbacks coach Bill Musgrave, league sources told Schefter.
The Seahawks confirmed Monday they have agreed to terms on a contract with Carroll to become their next head coach.
"We are excited to add Pete as our coach," said CEO Tod Leiweke in a statement. "He brings a great passion for winning and a positive attitude that is contagious. We now turn our full attention to the hiring process for a general manager.
"Our intended structure is for Pete and the new GM to work in a collaborative capacity on football matters."
Information from ESPN NFL Insider Adam Schefter, ESPN's Shelley Smith, ESPNLosAngeles.com's Ramona Shelburne and The Associated Press was used in this report.