The seven-figure offer was believed to have made on Tuesday and was swiftly turned down by PNE.
Cunningham is the third player North End have turned down significant bids for this summer, having rebuffed two offers from Birmingham City for Jordan Hugill and Derby County's interest in Callum Robinson.
Preston boss Alex Neil is determined to keep the core of his first-team squad in place - Cunningham, Hugill and Robinson are very much part of that - and North End will only consider offers for those on the fringes of Neil's plans.
That policy has the backing of club owner Trevor Hemmings who wants to help Neil develop the squad.
Cunningham, 26, has two years left on his Deepdale contract and has been a model of consistency since joining on a free transfer from Bristol City in July 2015.
On Monday, he was named in the Republic of Ireland's provisional 39-man squad for the forthcoming World Cup qualifiers against Georgia and Serbia.
Currently, PNE are waiting to discover the extent of a knee injury Cunningham suffered in the latter stages of Tuesday night's defeat to Derby County.
He hyper-flexed his right knee when landing awkwardly after a header and after initially playing on, had to leave the pitch for the last couple of minutes of stoppage-time.
Cardiff manager Neil Warnock is reportedly a long-time admirer of Cunningham and sees him as a potential replacement for Joe Bennett who is set to Join Fulham.