Medical staff took two hours to get Muamba breathing again and he is now being closely monitored in the Heart Attack Centre at The London Chest Hospital.
A joint statement from Bolton, Barts and The NHS Trust at 9.30am this morning read: "Fabrice Muamba's heart condition is stable, but he remains critically ill in intensive care at The London Chest Hospital.
"The family is grateful to the media for continuing to respect their privacy at this time."
Bolton manager Own Coyle and chairman Phil Gartside visited the hospital on Monday morning.
Coyle on Sunday led expressions of hope that the Muamba, a father of one, wins his fight for life.
Coyle, who accompanied Muamba to hospital, offered his personal thanks for the messages of support that continue to be received, both at the hospital and the Reebok Stadium.
"From our end, and on behalf of the family, we want to say a big thanks to everyone," said Coyle.
"We have been inundated by good messages, everyone praying for Fabrice which is very important. It has been a real source of strength for the family. We know how difficult a time it is for everyone.
"I can't really quantify it (the support). It has been incredible, not just from the football world, from everyone.
"It is great to be talented at football but it's more important to be a genuinely nice man and everyone recognises Fabrice is that.
"That has pulled everyone together.
"We certainly hope he comes through this. If he pulls through it would be marvellous, it really would."
Gartside added his own tribute: "I know Fabrice's family have been overwhelmed by the huge outpouring of kindness, from people visiting the hospital to leave cards to those sending messages and e-mails.
"The staff at the London Chest Hospital have been nothing short of exceptional and I would like to thank them all at this on-going critical time.
"I ask for everybody to keep Fabrice in their prayers and thoughts."
The Premier League agreed to Bolton's request for Tuesday's planned encounter with Aston Villa at Villa Park to be postponed.
No new date has been confirmed and it is still not clear when Saturday night's game will be played.
Bolton's next game is due to be in the Premier League, against local rivals and relegation scrappers Blackburn, at home on Saturday, although no decision will be made until much later in the week about whether that match can proceed.
After scoring in Chelsea's FA Cup win over Leicester at Stamford Bridge, Muamba's former Bolton team-mate Gary Cahill revealed a T-shirt with the message 'Pray 4 Muamba' printed on it.
Supporters at Wolves' home game with Manchester United combined in a minute's applause before kick-off.
Referee Howard Webb, whose prompt actions on Saturday, in addition to those of medical staff have received such praise, withdrew from fourth official duties at Wolves on the advice of a psychologist who works with the Professional Game Match Officials Limited (PMGMOL).