The 44-year-old suffered a cardiac arrest at the club's Enfield headquarters on Thursday morning. Ehiogu has Nigerian parents.
Emergency services had been scrambled to the scene at about 11.30am and Ehiogu was taken to hospital but died in the early hours of Friday morning.
The club said in a statement: "It is with immense sadness that we announce the passing of Ugo Ehiogu, our under-23 coach.
Head of Coaching and Player Development John McDermott added: "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. His younger brother Uzo Eziogu was a former captain in the British Military and a physiotherapist at the Defence Medical Rehabilitation Centre.
A Tottenham spokesman said the club sent its "deepest condolences" to Ehiogu's family, adding: "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 Hospotal for their care and support."
Ehiogu had been the club's Under-23s coach since 2014.
During his playing career he represented Aston Villa and Middlesbrough, and won four England caps.
Ehiogu began his career as a trainee at West Brom before he was taken to Villa by Ron Atkinson in 1991. He went on to feature in more than 300 matches for Villa and played in their FA Cup final defeat against Chelsea in 2000.
Ehiogu joined Middlesbrough for a then club-record £8million fee later that year, and in 2001 he scored his only goal for England in Sven Goran-Eriksson's first game in charge, a 3-0 friendly victory against Spain.
Knee injuries plagued the final years of Ehiogu's career at the Riverside Stadium.
He joined Leeds on loan in 2006, and also had short spells with Rangers and Sheffield United before he retired from football in 2009.
Ehiogu wrote his last message on his Twitter page three weeks ago, when he shared an impulsive good deed with his fans.
"Gave a homeless girl £10 last night in Dalston. She didn't ask or beg. Random impulsive act from me. Not gona lie. Felt good. #dosomethingkind".