The 61-year-old, who spent 12 years as a player at Spurs and also managed the London club, collapsed at the BT Sport studios.
On Sunday, a spokesman for Hoddle said he remained "in a serious condition" but was responding well to treatment.
Reports emerged on Monday that he had been moved out of intensive care and was able to talk to his family.
Tottenham tweeted a family statement on their official Twitter account at half-time of their clash with Manchester City at Wembley on Monday evening.
It read: "Glenn and his family are overwhelmed by the best wishes sent from Tottenham Hotspur staff, players, ex-players and most importantly all our tremendous supporters since he was taken ill on Saturday.
"It has been a huge comfort to have your support and memories to lift us all at such a difficult time, so thank you. Let's hope for a win tonight to make Hod smile. COYS."
Meanwhile, special constable Simon Daniels revealed how his "natural instinct" of first aid training helped former England manager Hoddle in the moments after he collapsed.
Daniels, a member of the production crew, gave Hoddle immediate medical attention and his swift actions were praised by both Hoddle's family and also Assistant Chief Constable of Thames Valley Police Nikki Ross, who said he had been "incredible".
Speaking to the Mirror, Daniels said his first aid training had just "kicked in" instinctively as he went to help.
"The live programme had just come off air. I was in the sound control room. Glenn was there with other presenters. I heard a bit of commotion," Daniels said.
"My natural instinct was to help. Glenn had collapsed and was clearly unwell.
"I did the standard ABC [airway, breathing, circulation] checks. I immediately started CPR, a defibrillator was brought to me. I continued CPR and used the defib.
"The paramedics then turned up and I continued CPR until he was taken to hospital."
Daniels added: "Our number-one priority is always preservation of life, and so my training just kicked in.
"It wasn't stressful, but it was hard work. It wasn't a matter of feeling stressed it was purely a question of his wellbeing.
"I don't see myself as a hero, but I am very touched by all of the comments, both from Glenn's family and the wider football and television community.
"My thoughts are with Glenn Hoddle and his family and I hope he continues his recovery. I look forward to seeing him make a full recovery.
"I would also like to praise the medical professionals, the ambulance service and doctors who worked on him."
Daniels had also replied to a Twitter post from BT Sport presenter Robbie Savage which called him a "hero". The sound engineer maintaining he "just did what was necessary to give him a fighting chance" and his training as a special constable had "paid off."
The football community was quick to support Hoddle, who won 53 caps for England.
England manager Gareth 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.
"Everybody across the game is right behind one of English football's greats and we are willing him to pull through."
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