Fernando Alonso sustained a fracture of his upper jaw in his cycling accident on Thursday, his Alpine team say.
The 39-year-old two-time world champion had a "successful corrective operation" and will remain in hospital for observation for 48 hours.
The team added: "The medical team are satisfied with his progress. After a few days of complete rest, he will be able to progressively resume training.
"We expect him to be fully operational to [prepare] for the season."
Pre-season testing is to be held in Bahrain from 12-14 March, before the first race at the same track on 26-28 March.
The Spaniard was involved in a collision with a car near his home in Lugano, Switzerland, while out for a training ride.
A report by police in Lugano, where Alonso lives, said the accident happened as a car turned left, across oncoming traffic, into a supermarket car park.
The driver of the car was a 42-year-old Swiss who lives in Lugano.
The police report said: "For reasons the police investigation will have to establish, while [the driver was] making a left turn manoeuvre to enter a supermarket parking lot, there was a collision with a 39-year-old Spanish citizen, who was riding his bicycle and passing the column of stationary vehicles in the opposite direction on the right.
"The collision occurred against the right-hand side of the car."
Alonso was attended to by police officers and Green Cross paramedics before being transported to hospital.
Alonso made his debut in F1 in 2001, and in a 17-year career for the Minardi, Renault, McLaren and Ferrari teams established himself as one of the greatest racing drivers of all time.
A winner of 32 grands prix - sixth in the all-time list - and world champion in 2005 and 2006, he left F1 at the end of 2018 after five seasons with an uncompetitive McLaren to pursue other interests in racing.
In the intervening two years, he added a second consecutive victory in the Le Mans 24 Hours and the World Endurance Championship title with Toyota, twice competed without success in the Indianapolis 500 and took part in the Dakar Rally.
He has said he is returning to F1 because he loves driving and competition, and has hopes that Alpine, the former Renault team, will be competitive when new technical rules are introduced in 2022.
The team scored three podiums last year, two for Daniel Ricciardo, now with McLaren, and one for Esteban Ocon, who will be Alonso's team-mate in 2021.
Alonso has already missed the start of a season once through injury.
When he returned to McLaren in 2015, he suffered concussion in an accident in pre-season testing and had to sit out the first race in Australia while recovering.
- Standing Up To Infertility: Rhod Gilbert with a frank, funny and personally revealing look at the condition
- Scarlett Moffatt Wants To Believe: What is the theory behind ghost sightings and does it prove anything?