Team principal Fred Vasseur will oversee the chassis area as an interim measure.
Cardile has spent almost two decades at Ferrari, who are second in the championship after winning two of 12 races so far but are in danger of being overhauled by resurgent McLaren who are only seven points behind.
Media reports in Italy have linked Cardile to rivals Aston Martin, whose double world champion Fernando Alonso was previously at Ferrari.
Loic Serra is due to join Ferrari from Mercedes in October as head of chassis performance engineering and would have reported to Cardile.
The sport's most successful and glamorous team are also attempting to recruit star designer Adrian Newey, who has announced his departure from Red Bull without saying where he might go.
Aston Martin have also made Newey an offer, according to F1 media and British outlets.
Canadian driver Lance Stroll, son of Aston Martin team owner Lawrence, said recently that Newey was "someone that everyone would love to have in their team colours".
Ferrari have seven-time world champion Lewis Hamilton, the winner of his home British Grand Prix last Sunday, joining from Mercedes next season.
The Italian team struggled at Silverstone with Carlos Sainz finishing a solid fifth, with a bonus point for fastest lap, but Charles Leclerc languishing in 14th - his second pointless race in a row.
Ferrari introduced an upgrade package at last month's Spanish Grand Prix but problems forced a return to the older specification.
"We are still on the Imola car, which is already two months ago, and in F1 if you are two months standing still the others are going to bite you," Sainz told reporters.
