Albon is 19th in the drivers' championship
The 26-year-old had appendix surgery on Saturday during the Italian GP race weekend but suffered "post-operative anaesthetic complications" and was put on a ventilator in Italy.
Thai Albon is now back at his home in Monaco and "feels pretty good".
"The goal is to be ready for Singapore," said Albon.
The Singapore race weekend at the Marina Bay Street Circuit begins on 30 September.
Speaking in a video on Twitter, Albon thanked doctors for doing "an amazing job", adding: "I'm very grateful that they got me in good health".
Of his aim to be fit for the next race in Singapore, he said: "It's going to be tough, it's one of the toughest races that we go to. It won't be easy but let's aim high."
Albon was replaced by reserve driver Nyck de Vries for Sunday's Italian Grand Prix, which was won by Max Verstappen.
-- Courtesy of BBC Sport