Hoddle was taken ill on Saturday as he prepared to go on air for BT’s weekly show BT Sport Score.
It has since been revealed that the former England manager suffered a heart attack, before quickly being rushed to hospital.
Hoddle’s condition was described as “serious” on Saturday but he was receiving specialist treatment and said to be “responding well”.
Simon Davies, a member of the production crew and a special constable, was present when Hoddle collapsed and offered him immediate medical attention.
A spokesman of Hoddle released a statement on Sunday, saying: “Glenn remains in a serious condition in hospital after suffering a heart attack yesterday. He continues to respond well to treatment.
“The family are grateful to everyone in the football family – and beyond – that have sent kind messages of support, they are very much appreciated.
“In particular, Glenn and his family would like to publicly thank the BT Sport staff that treated him immediately on set following his collapse.
“Glenn is now in the care of the professional NHS medical services, who have also been exemplary in helping him and the family during the last 24 hours.
“Doctors have advised the most important thing for Glenn is time to rest. Therefore, his family have reiterated the request for their privacy to be respected during this period.”
Daniels, a special sergeant with Thames Valley Police, says he took the necessary measures to give Hoddle a “fighting chance”.
“So pleased that Glenn is now getting the best care possible in hospital,” Daniels tweeted.
“Just did what was necessary to give him a fighting chance. Thoughts are with his family. It was my training as a Special Constable that paid off.”
Hoddle played for and managed Chelsea, Tottenham and England, and is regarded as one of the greatest players of his generation.
Soon after the news broke, Three Lions boss Gareth Southgate and FA chairman Greg Clarke sent their well-wishes.
“My thoughts and prayers are with Glenn Hoddle and those closest to him following today’s distressing news,” Southgate said.
“Everybody across the game is right behind one of English football’s greats and we are willing him to pull through.”
Clarke added: “All of us at Wembley and St George’s Park are thinking of Glenn and hoping for positive news.
“Our thoughts are also with his family and friends at what will be an extremely difficult time for everyone who knows him.”