Bogar, 45, has been with the Red Sox since 2009, and his current contract expires Wednesday.
New Red Sox manager John Farrell is expected to name Torey Lovullo as his bench coach. Lovullo managed Triple-A Pawtucket in 2010 before joining Farrell in Toronto as its first-base coach.
Bogar was Boston's first-base coach in 2009, then served as the third-base coach in 2010-11 before being named bench coach last season under former manager Bobby Valentine.
Bogar is well-respected by Red Sox players.
"To me, Bogey was that calming voice that was always thinking baseball," Red Sox second baseman Dustin Pedroia told ESPNBoston.com on Wednesday. "He was trying to put us in the best situation to succeed, whether that was baserunning, defensive positioning or just mentioning to calm down at the plate. And his timing was always right on. He's a very smart, no-nonsense guy, which you appreciate."
Bogar was recently offered the bench coach job with the Houston Astros, but he turned it down.