The 36-year-old centre-back became a free agent recently, after leaving Premier League champions Chelsea at the end of the season, and Redknapp is keen on bringing his experience to City.
“We’ve made him a good offer and we have done the best we can,” Redknapp told TalkSport.
“It’s up to John now, but we would love him at Birmingham.”
Terry won 13 major trophies with Chelsea and made 717 appearances for the club, but only featured in 14 games during the 2016/17 campaign.
Birmingham finished 19th in the Championship last season. The last time they competed in the top flight was in 2011.