Hoddle collapsed at the studios and was rushed to hospital with BT cancelling one of their football shows.
BT Sport presenter Jake Humphrey tweeted that Hoddle "was taken seriously ill". We're right with you, Glenn'.
Later in the day, Humphrey provided an update on Hoddle's condition, saying: "The condition is serious but Glenn is now receiving specialist treatment and responding well".
Swindon Town added: "We would like to take this moment to send our support and best wishes to Glenn Hoddle".
Our thoughts are with him and his entire family.
Hoddle was due to be a pundit for West Ham's trip to Leicester's King Power Stadium on Saturday evening.
Southgate, who played under Hoddle for England, said: "My thoughts and prayers are with Glenn Hoddle and those closest to him following today's distressing news".
Hoddle then had spells managing Swindon and Chelsea before taking charge of England between 1996 and 1999, giving David Beckham his worldwide debut in 1996.
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During his career he also played for Monaco, Swindon and Chelsea.
Is Glenn one of your all time footballing heroes?
Hoddle, now a TV soccer pundit, is widely seen as one of the most skillful footballers of his generation. You won't meet a nicer guy than Glenn. "All our thoughts are with Glenn Hoddle and his family".
"He's got a strong character and I'm sure he'll battle through it".
His former employers expressed their concern at the information, posting on the official England Twitter account: "Worrying news".
He lost his job the following year after controversial remarks he made suggesting disabled people were paying the price for sins in past lives in an interview with The Times - the remarks all the stranger as he was an ardent fundraiser for disabled peoples charities.
Former England goalkeeper Ray Clemence, who was a goalkeeping coach during Hoddle's England reign, urged him to "stay strong". "Great manager and a great man".
Tottenham Hotspur, the club where Hoddle starred as a super-talented midfielder winning two FA Cups and a Uefa Cup and went on to manage, tweeted their support.