Birmingham's improved bid is worth up to £5m but the cash down-payment for Johnson is thought to be less than half of the headline figure with the deal made up in add-ons should the Blues remain in the Premier League.
Johnson is one of a number of Cardiff players who have been targeted by Premier League clubs.
Captain and Wales midfielder Joe Ledley, who has one year left on his Ninian Park deal, has been linked with moves to Everton, West Ham United and Stoke City.
And 23-goal top-scorer and striker Ross McCormack, who last week publicly stated his desire to play in the Premier League, has been linked with a number of top-flight clubs.
But Cardiff City have confirmed that the club has received no bids for any of their other players.
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