Blues manager Harry Redknapp has made no secret of his desire to sign the 28-year-old this summer and submitted an improved offer last week after his initial proposal - a joint deal also involving Joe Bryan - was turned down.
Robins owner Steve Lansdown is thought to be seeking £10m for Flint, and Brum's latest bid has not come close to the valuation.
"We have had two solo bids from Birmingham. Neither reached the acceptable valuation for Aden," City manager Lee Johnson told the club's Twitter feed.
"The owner has gone on record to say we don't need to sell. It has to be right for our club at the right valuation."
Flint has racked up more than 200 appearances for Bristol in all competitions since joining from Swindon Town four years ago.