"We are expecting him to turn up on the show on Sunday... He doesn't want to leave his acts to perform without him there.
"Provided he continues to improve, he will do everything he can to be there."
Cowell was prescribed morphine for his pain and will continue to undergo tests this week to determine why he passed out.
The Britain's Got Talent star was looking after his three-year-old son, Eric, when he fainted, but his girlfriend Lauren Silverman has now returned to London from New York to be with them while he recovers.
A source previously said: "Simon was at his home alone with Eric, apart from his security team and nanny.
"Lauren's son Adam lives in New York and it was her week to look after him there.
"But she was distraught to be woken up by the news and immediately rushed to the airport to get back home to be with Simon. She is upset and worried.
"Luckily there was a nanny at the house who was able to look after Eric."
It was previously revealed medics feared Cowell had broken his neck when he fell.
A source said: "Simon had woken up and wasn't feeling right. He was dizzy so went downstairs to fetch some warm milk.
"He fainted when he was climbing back up the stairs to the first floor. His security team heard a loud thump and knew he'd gone down.
"They immediately called emergency services and an ambulance rushed to the house.
"But he couldn't be moved for some time because he felt a pain in his spine so they were worried there could be some type of much more serious problem like a broken back."