Redknapp’s most recent managerial post was with Jordan. He last worked in English football with Derby County as an advisor.
Zola stood down on Monday, declaring “I sacked myself”.
The 50-year-old was a controversial appointment by the club’s Chinese owners after they sacked Gary Rowett, who had Birmingham well placed to challenge for a place in the play-offs, in December.
Redknapp’s first game as Blues’ boss is Saturday’s Birmingham derby at Aston Villa.
His last spell in the Championship saw Redknapp guide QPR to promotion. Redknapp has also managed two other London clubs in West Ham, for whom he played, and Tottenham Hotspur.
He has also been in charge of three south coast clubs, starting his managerial career at Bournemouth and later twice taking charge of Portsmouth either side of an ill-fated stint at their bitter rivals Southampton.
It was with Portsmouth that Redknapp enjoyed arguably his greatest success when they won the 2008 FA Cup.
“I sacked myself,” he explained. “I decided to give in my resignation. I am sorry because I came to Birmingham with huge expectations.
“Unfortunately the results have not been good and I take full responsibility. It is not that I like quitting, but Birmingham deserves better.”
Zola, in his third management spell in England after stints at West Ham and Watford, added: “If I feel I cannot help the players, why stay? If I cannot help the team, it is better I leave and let someone else do that.
“I feel very bad and very sorry. We worked with a lot of meaning, but unfortunately it didn’t produce the results. It is all very disappointing.”