The ex-West Ham player and Manager succeeds former West Ham manager Gianfranco Zola, who resigned on Monday following a 2-0 home defeat by Burton Albion.
“Birmingham are a proper football club but they are in a precarious position,” 70-year-old Redknapp told TalkSport.
Redknapp, who will be assisted by former Bristol City boss Steve Cotterill, told Talksport: “I got a phone call last night at 7pm from the people of Birmingham.
“I drove to London and had a 10-15 minute meeting with them and said ‘sure I’ll come and do it’. My wife said to me ‘are you mad or what?’ but I get fed up sitting around doing nothing.
“It’s a real challenge. I’ll come and live up there until the end of the season and if I keep them up then I’ll sit down and talk about next season. It’s not really a risk. They have won two out 22. I haven’t got a magic wand. I’m not going to turn them into Real Madrid. We need a win.”