Racing Bulls are currently heading for the team's best finish to a season, in fifth place and 39 points clear of sixth-placed Alpine, with four races remaining including Sunday's Sao Paulo Grand Prix in Brazil.
Permane joined as racing director in January 2024 after 34 years at Enstone with the team that is now Renault-owned Alpine. He took the principal position in July 2025 after Laurent Mekies took over from Christian Horner at Red Bull.
The Italy-based team finished sixth overall last year.
Red Bull last month extended Mekies' contract to the end of 2031.
Racing Bulls have yet to confirm their 2027 driver lineup. British rookie Arvid Lindblad and New Zealander Liam Lawson are the current pairing.