Alpine issued a statement on Thursday evening saying that Alonso was "conscious and well in himself", pending further examinations.
Sources have indicated to Motorsport.com that Alonso is now set to be moved to a hospital in Bern, where he will undergo surgery for some broken teeth.
It is understood that Alonso came off his bike while braking hard to avoid an impact with a car while out cycling on the road.
The early indications are that Alonso has not sustained any fractures in the accident, but he is set to undergo further medical examinations on Friday.
The Alpine team is set to offer further updates on his condition later today.
Alonso is set to make his F1 comeback this year after two seasons out of the series, with his last grand prix start coming for McLaren at the end of 2018.
Since then, Alonso has won the 24 Hours of Le Mans twice for Toyota, along with one World Endurance Championship title, and has also competed in the Indianapolis 500 and the Dakar Rally.
Alonso announced last summer that he would be returning to F1 with Renault - which has since been rebranded as Alpine - to mark his third stint with the Enstone-based squad.
Alonso raced for Renault between 2003 and 2006, claiming the drivers' world championship in 2005 and 2006, before spending one season at McLaren. A return to Renault in 2008 yielded a further two victories before he switched to Ferrari for 2010.
A second McLaren stint followed from 2015 to 2018, but the team's dip in on-track fortunes left Alonso increasingly frustrated, prompting him to explore avenues outside of F1.
This led to Alonso making his Indy 500 debut in 2017 as he set his sights on the Triple Crown of Motorsport, looking to add victories at Indianapolis and Le Mans to his Monaco wins.
He is due to make his official on-track return ahead of the new season in pre-season testing, which begins in Bahrain on 12 March.