The former England boss is said to be 'responding well' to treatment after he was rushed to hospital from the studios in Stratford, East London.
The channel's Score programme was cancelled just minutes before it was due to go live on air at 2.45pm as paramedics were called to the scene.
A spokesman confirmed the ex-England boss is surrounded by his loved-ones and is receiving specialist care at a hospital in London.
He said: “Earlier today Glenn collapsed at the BT Sport studios. He received immediate medical attention on set and was subsequently transferred by paramedics to hospital in London.
“The condition is serious but Glenn is currently receiving specialist treatment and responding well.
“Glenn’s family are with him and would like to thank everyone that has sent their support. They have also asked for their privacy to be respected at this time.”
Fellow show hosts Gary Lineker and Jake Humphrey have expressed their concern for the legendary footballer who is widely considered to be one of Tottenham Hotspur's greatest ever players.
A spokesman said: "Earlier today Glenn Hoddle was taken ill at the BT Sport studio.
"Glenn received medical attention on site and he was then taken to hospital. All our thoughts are with Glenn and his family."
Host Humphrey offered an update on the legendary footballer during the channel's coverage of Leicester vs West Ham.
He said: "The good news is that he is now at hospital getting the very best treatment possible.
“Whatever Glenn Hoddle is to you, whether it’s one of the greatest footballers this country has ever produced, a wonderful colleague, a much-loved family member, the outpouring of love and support today has been quite something to behold.
“And of course all of us at BT Sport absolutely add to that.
“We are standing alongside you Glenn and sending you are very best for a speedy recovery.”
On Twitter, Lineker revealed: "Hugely worrying news that Glenn Hoddle collapsed at the BT studios earlier. Thoughts are with him and his family. Come on Glenn."
Earlier, Humphrey tweeted: "Our friend & colleague Glenn Hoddle was taken seriously ill at the BT Sport studio this morning.
"For that reason there will be no @btsportscore – but we will be live as usual for the 5.30 ko at Leicester.
"Everyone of us is right with you Glenn, sending love and strength."
The Tottenham legend earned 53 caps for England after starting his career at Spurs, where he stayed for 12 years.
He won the FA Cup and Uefa Cup with the North London club and also played for French side Monaco before finishing his career as player-manager of Swindon and Chelsea.
Following his stint in charge of Chelsea he succeeded Terry Venables as England manager in 1996 – leading the Three Lions to the 1998 World Cup in France.
Hoddle's England side, which boasted starlets such as Michael Owen, Paul Scholes and David Beckham, were knocked out on penalties by Argentina in the second round of the tournament.
His reign ended in controversy after he suggested disabled people were being made to pay for sins in a past life.
He later admitted making a "serious error of judgement" over the comment.
Since quitting management in 2004, he has been a regular pundit and commentator.
He was a pundit for BT Sport’s Europa League coverage of Arsenal vs Sporting on Thursday.
Support has flooded in from the football world as news of his collapse spread today.
Chelsea tweeted: “Our thoughts are with former Blues boss Glenn Hoddle, who has been taken seriously ill on his 61st birthday. We’re right with you, Glenn.”
Ex-England star Ian Wright wrote: "Hang in Gaffer".
Tottenham said: "Everybody at the Club sends their best wishes to @GlennHoddle for a full and speedy recovery after he was taken ill this morning.”
While England tweeted: "Worrying news. Get well soon, @GlennHoddle – we’re all behind you."
Football Focus presenter Dan Walker said: "Very worrying to hear about Glenn Hoddle collapsing and being taken to hospital. Really hope he’s ok."
Hoddle has been married to second wife Vanessa Colbourn since 2000. and has three children – Zara Marie, Jamie Glenn and Zoe Ann.
He was struck by tragedy in 2008 when his brother Carl died aged just 40 from a brain aneurysm.
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