The approach was made on Tuesday but was quickly turned down by Preston.
Cunningham was seen as a possible replacement for Joe Bennett, who has been in talks with Fulham after they triggered his release clause.
But Bennett may now stay with the Bluebirds and is weighing up the offer, with the player considering staying at the Cardiff City Stadium.
Republic of Ireland international Cunningham has made more than 50 appearances for Preston.
Bluebirds manager Neil Warnock confirmed after his side's 2-0 win over Sheffield United that Fulham had made an approach for Bennett.
"There is a clause in his contract ... and if they trigger that clause there's not a lot I can do," he said.
Warnock would not confirm if Cardiff had made specific offers for players, but added: "There's a few the media have not mentioned that I've been talking to. You'll just have to wait and see."
Cardiff City have had a bid for Preston North End defender Greg Cunningham turned down.