May led the Bluebirds to the Division Three title in 1993 and a famous FA Cup fourth-round win over Manchester City at Ninian Park the following year during a four-year spell at the club. The former defender, who died on Saturday, was born in Epping, but spent most of his playing career in Wales with Wrexham and then Swansea.
Cardiff chief executive Alan Whiteley told the club's official website: "On behalf of everyone at the club I would like to pass on our sincere condolences to Eddie May's family at this sad time."
He added: "We will be honoured this week to collectively show our respects and appreciation to a man who will forever and rightly be remembered as an integral part of modern Cardiff City history. He will be sadly missed by all."
Swansea and Wrexham also paid tribute to their former player on their official websites. Wrexham announced May had died in his sleep.
May, who established himself as a player at Southend, spent eight years at Wrexham before finishing his career with a three-year stint at Swansea.
His managerial career took him across the globe.
Aside from his Cardiff exploits, he had stints at Newport County, Brentford and Torquay in the English leagues, Barry Town, Haverfordwest, Llanelli and Porthcawl in the Welsh and also managed abroad in Iceland, Ireland, Kenya, Norway, Qatar, Pakistan, Saudi Arabia, South Africa, Uganda and Zimbabwe.
Cardiff will hold an official tribute to May on Saturday ahead of the home match against Leeds, where there will be a minute's applause.
The players will also wear black armbands against Derby on Tuesday.