Today is his 61st birthday.
Earlier on Saturday Hoddle had appeared on Saturday Morning Sav with Robbie Savage, Paul Ince, Steve Sidwell and Harry Redknapp.
A statement from Hoddle's family said: "The family are grateful to everyone in the football family - and beyond - that have sent kind messages of support, they are very much appreciated".
"Earlier today Glenn collapsed at the BT Sport studios - he received immediate medical attention on set and was transferred by paramedics to a hospital in London. Our thoughts are also with his family and friends at what will be an extremely hard time for everyone who knows him". "Therefore, his family have reiterated the request for their privacy to be respected during this period".
Hoddle, who also works for ITV Sport on England games, was one of English football's top players during the 1980s, an elegant midfielder who won 53 England caps before managing his country from 1996 to 1999.
He was taken ill at the studios of BT Sport, who consequently cancelled their usual Saturday scores and results show.
Glenn Hoddle is in hospital after being taken seriously ill.
A source close to the family told a former teammate that Hoddle had been "close to death" and was saved by the quick action of an employee at BT Sport who knew how to use a defibrillator. "Our thoughts are also with his family and friends at what will be an extremely hard time for everyone who knows him". He played for Tottenham, Chelsea and Monaco.
Soon after the news became public, the stream of support, flowing from so many corners of the football landscape, testified to the popularity of a man who both played for and managed England. Thoughts are with his family. "We're right with you, Glenn".
The England national team sent their well wishes to Hoddle, tweeting: "Worrying news".
We would like to send our best wishes to Glenn and all his family at this hard time. "All our thoughts are with Glenn Hoddle and his family".