The former Tottenham and Portsmouth boss then signed a one-year deal in May to stay on as Blues boss.
"Lee Carsley, who is well respected on the coaching staff and in temporary coach, is priced at 5/1".
Saturday's 3-1 home loss to Preston was the final straw and Redknapp finds himself out of work once again.
Birmingham have lost seven of their 10 games across all competitions this season, including each of their last six.
'At least' four dead in United Kingdom motorway pile-up
He was one of the first at the scene along with a passing GP and other medics to help an injured woman who was in one of the cars. Highways England said the "early expectation" is that the M5 "will remain closed in both directions late into this evening".
But despite investing heavily in new players over the last few months, Birmingham have made a decision to axe the 70-year-old after six successive defeats, which sees them sitting second from bottom of the Championship.
Port Vale's CEO, Colin Garlick, has revealed that David Kelly and Chris Morgan have been placed in charge of the first team while the club's search for a new manager, with both men joining the likes of Justin Edinburgh, Neil Redfearn and Neil Aspin in the betting to replace Brown.
"Fans deserve better and that was my motivation to carry on this season".
Redknapp, who signed 14 players during a major summer overhaul, was in charge at St Andrews for 13 games. "I would love to be the manager to bring some success to Birmingham".
"I didn't get the players in I was after in pre-season and suddenly I have got a load of players in", he told the Birmingham Mail. He takes the job in a caretaker role.
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