The Spurs Under-23s coach collapsed on the club's academy training pitches and received immediate attention from the club's medical staff.
Emergency services arrived at the scene at approximately 11.30am and Ehiogu was taken to hospital but died in the early hours of Friday morning.
The club issued a statement, which read: "It is with immense sadness that we announce the passing of Ugo Ehiogu, our Under-23 coach.
"Ugo received immediate treatment after suffering a cardiac arrest at the Training Centre yesterday before being transferred to hospital, where he passed away in the early hours of this morning.
"The club sends its deepest condolences to Ugo's family. Gemma has specifically asked that the family are given privacy at this difficult time.
"We should also like to place on record our thanks to all the medical professionals including those at North Middlesex University Hospital, Royal Brompton Hospital and Harefield Hospital for their care and support."
John McDermott, Tottenham's head of coaching and development, said: "Words cannot express the shock and sadness that we all feel at the club. Ugo's immense presence will be irreplaceable. Our heartfelt sympathies go out to his wife Gemma and his family."
Ehiogu made over 200 appearances for Aston Villa between 1991 and 2000, and then spent seven years at Middlesbrough, winning the League Cup with Villa in 1994 and 1996, and also with Boro in 2004.
The centre-back also played for West Bromwich Albion, Leeds United, Rangers and Sheffield United, and won four England senior caps before retiring in 2009.