The midfielder was rushed to hospital on Saturday evening after collapsing during the first half of his side's FA Cup sixth-round tie against Tottenham
The club also confirmed the 23-year-old's heart had started working at the London Chest Hospital following prolonged resuscitation after he collapsed in the 41st minute at White Hart Lane.
A statement issued jointly by Bolton and the London Chest Hospital read: "Fabrice Muamba remains in a critical condition in intensive care in the Heart Attack Centre at The London Chest Hospital.
"Fabrice was admitted to the hospital yesterday evening after collapsing at White Hart Lane, where he sustained a cardiac arrest during the FA Cup quarter-final against Tottenham Hotspur.
"Fabrice received prolonged resuscitation at the ground and on route to The London Chest Hospital, where his heart eventually started working.
"As is normal medical practice, Fabrice remains anaesthetised in intensive care and will be for at least 24 hours.
"His condition continues to be closely monitored by the cardiac specialists at the hospital. No further updates are planned at this time."
Bolton manager Owen Coyle, who accompanied the midfielder to hospital alongside striker Kevin Davies and club chairman Phil Gartside, thanked the wave of support for Muamba.
"Fabrice's family have asked me to pass on their thanks for the many, many kind messages of support from not only Bolton fans but also fans from clubs across the country and abroad," he said.
"All our thoughts and prayers are for Fabrice and his family. The family would also like to thank the media for respecting their privacy at this time."
Earlier on Sunday morning, Coyle had given an update on Muamba's condition.
"Fabrice is critically ill. The next 24 hours are going to be absolutely crucial," he said.
"We've obviously been inundated with people wishing him well and we hope that if everybody can pray strongly tonight that Fabrice is able to recover.
"It's very serious. There's not getting away from that. He's critically ill and God willing he makes it through."