Presenter Mark Pougatch said the station's football results show was cancelled as a result.
The former Spurs midfielder earned 53 England caps and was considered to be one of the players of his generation.
BT presenter Jake Humphrey said Hoddle was "taken seriously ill at the BT Sport studio this morning".
He added on social media: "Every one of us is right with you, Glenn, sending love and strength."
Gary Lineker, the Match of the Day presenter and former Tottenham and England striker, posted on social media: "Hugely worrying news that Glenn Hoddle collapsed at the BT studios earlier. Thoughts are with him and his family. Come on, Glenn."
As a player, Hoddle won the FA Cup and Uefa Cup with Tottenham and also had spells with Monaco, Swindon and Chelsea.
Hoddle became player-manager at Swindon and Chelsea, and is credited with turning the west London side into a cosmopolitan outfit by signing Netherlands great Ruud Gullit, who went on to replace him as boss at Stamford Bridge.
From there he managed England for two and half years, taking them to the 1998 World Cup, but his reign ended in controversy when he admitted a "serious error of judgement" after suggesting disabled people were being made to pay for the sins of past lives.
He went on to manage Southampton, Tottenham and Wolves, before continuing coaching and his career as a TV pundit.
Former Spurs team-mates were quick to send their support to Hoddle on social media.
Ex-Argentina midfielder Ossie Ardiles said: "Get well soon Tommy! Our thoughts and prayers with him and his family."
Former goalkeeper Ray Clemence urged Hoddle to "stay strong", while Chris Waddle, another ex-England international said: "Get well soon, Glenn. Our thoughts are with you."
Brighton boss Chris Hughton, twice an FA Cup winner with Spurs alongside Hoddle, said: "Glenn is a friend, I have known him for a long time back to my Spurs youth days. He was a model professional, the one we looked up to and we wish him all the best."
Ex-Spurs defender Graham Roberts said: "Thinking of the greatest player I have ever played with, we know you will be fine Glenn, thinking of you mate."