The Bluebirds have matched an offer from Celtic; though the Scottish champions are reportedly in advanced talks with the former Leicester man.
As part of Celtic's offer, 22-year-old midfielder Ryan Christie could be on the move in the opposite direction to Pittodrie with Aberdeen next season.
Cardiff made two failed attempts to sign Hayes in January, and the Republic of Ireland international has since signed a new deal with Aberdeen until 2019.
Hayes scored 11 goals in 43 appearances for Aberdeen last season as they finished second in the league, behind Brendan Rodgers’ dominant Celtic side.
The 29-year-old came through Reading’s academy, but failed to make a first team appearance for the Royals. He signed for Leicester City in 2007, but faced further struggles for regular game time.
In 2009, after loan spells at Macclesfield and Northampton, he joined Inverness Caledonian Thistle on a free.
After three years in the Tulloch Caledonian Stadium he moved to Aberdeen, having amassed 30 goals in his 200 appearances for the club so far.