The outfit formerly led by Christian Horner claimed the Mercedes star had braked erratically under safety car conditions and 'displayed unsportsmanlike intent' for his team radio complaints about Max Verstappen.
The current sporting regulations mean F1 teams must pay a refundable deposit of €2,000 for a protest or a right of review, with an appeal priced at €6,000, which is only refunded if successful.
The Canadian GP protest was rejected by the FIA and Red Bull's protest deposit was forfeited, leading to an ongoing debate in the paddock over whether the protest fee should be increased in order to deter teams from making protests that did 'not warrant being reported to the stewards'.
Following this week's meeting, a report was released by the FIA that read: "It was agreed that the deposit fees for protests, appeals and right of reviews should be evaluated with a view to those fees being adjusted. The introduction of a fee for investigations was also discussed."
Red Bull attempted to have George Russell's Canadian GP win stripped by the FIA
Red Bull rivals head for F1 boost
A lot has changed since June's race in Canada but Red Bull's F1 rivals will gladly welcome the FIA's willingness to discuss the increase of protest fees.
After the process between Red Bull and Mercedes had been dealt with by the FIA in Montreal, Silver Arrows team boss Toto Wolff labelled the actions of their rival as 'petty' and 'embarrassing'.
Wolff then suggested a change needed to be made to prevent similar incidents in the future, saying: "They come up with some weird clauses, what they call clauses. I guess the FIA needs to look at that because it's so far-fetched it was rejected."
The message from Mercedes' team principal echoed one which had already been issued by fellow rival boss Zak Brown earlier this season, once again as a result of the actions of Red Bull.
McLaren previously faced allegations that they were using water to cool tyres during races, and although the papaya outfit were cleared over the matter, those allegations were understood to have come from Red Bull.
The incident led McLaren CEO Brown to suggest a larger deterrent was needed from the FIA, declaring back in May: "There's a proper way to protest a team at the end of the race, and you have to make it formal, disclose where it comes from, put some money down.
"I think that process should be extended to all allegations to stop the frivolous allegations which are intended only to be a distraction."