Albon was admitted to intensive care earlier this month when he suffered respiratory failure after what would otherwise have been a routine operation. He was replaced at the Italian Grand Prix by stand-in Nyck de Vries, who finished ninth.
Albon previously said he would have a clear idea by the end of Friday's two practice sessions if his body is up to the task, and the driver said he believes he is fit to race.
"It was on the limit, but if I think I am able to do it, then I want to be in. And now I know I am," Albon told the BBC.
The race in Singapore is the most physically demanding on the F1 calendar, and Albon's training included some karting to see how his body was recovering from his stint at hospital.
Speaking at a news conference on Thursday, Albon said he was "lucky" to miss just one race.
"In terms of setbacks, it's a small one, really," Albon said. "I've missed out on a race, I've been very lucky. I've had very good doctors around me in Italy to get me back to a good place.
"I feel very fortunate. I only missed one race so it's not a big deal."