Brivio joined Alpine at the start of the 2021 season as its racing director in F1, having the previous year guided Suzuki to its first MotoGP world championship in 20 years with Joan Mir.
The Italian was moved out of the F1 team for the 2022 season, however, to head its young driver programme as director of racing expansion projects.
Brivio's current contract with Alpine did extend into next year, but he and the company have agreed mutually to part ways at the end of 2023.
This move comes just weeks after a report in Speedweek stated he was being lined up to replace Alberto Puig as Repsol Honda's team boss in MotoGP.
Honda has been mired in a difficult period in MotoGP of late, with it losing Marc Marquez to Gresini Ducati for 2024, while internally it has undergone a number of senior personnel changes.
However, Motorsport.com understands Brivio does not have an offer from Honda to join for 2024.
"It has been a proud chapter in my motorsport career to be involved in Formula 1 with Alpine," Brivio said.
"I would like to thank Alpine for the opportunity to experience Formula 1 which was my desire and also for the chance to pass on some of my experience in motorsport onto its young drivers in the Alpine Academy.
"I wish the team and the Academy the best in the future and I'm sure we will see many young drivers go on to achieve fantastic things in their careers.
"To play a small part in some of that success will certainly be something I will cherish.
"I am grateful to Alpine for accommodating my wish to pursue other opportunities which may (and I hope that they will) arise in the future."
Brivio leaving Alpine marks another significant personnel reshuffle within the team in 2023, which included CEO Laurent Rossi being moved aside to be replaced by Philippe Krief.
Team boss Otmar Szafnauer and sporting director Alan Permane were also ousted from their roles, while chief technical officer Pat Fry left for Williams.
This all came as the Enstone-based Alpine team suffered its worst finish in the F1 constructors' table since 2017, having slid from fourth in 2022 to sixth.
Alpine has confirmed Julian Rouse will continue to oversee its Academy programme.
At this stage, it is unclear what Brivio's next career move will be.
Prior to helming Suzuki from 2013 ahead of its return to MotoGP full-time in 2015, Brivio acted as team manager at Yamaha from 2004 to 2010.
Brivio was instrumental in signing Valentino Rossi from Honda to join Yamaha from 2004, which ultimately resulted in four world titles for the marque in 2004, 2005, 2008 and 2009.
Rossi's arrival also ended Yamaha's 12-year title drought and returned it to race-winning ways in 2004 having gone without a single victory in 2003 – something that wouldn't happen again until this year.
