Johnson cost City £275,000 from Wycombe three years ago and is one of the prize asset's in the Bluebirds squad.
"We put in a bid for £3.5m two weeks ago for Johnson who we rate highly," a source at St Andrews told The Echo.
"But we were told he is not for sale at any price. It's a starting bid but we have had little encouragement coming back from Cardiff.
"We were told that the price was well below what they rated the player.
"The danger is that we have other players to check out at other clubs in the coming days.
"They might be more positive in their feedback in giving someone the chance to play Premier League football."