Dr Helmut Marko had what he described as a "very important" meeting with Honda last Monday.
The Dutch publication Formule 1 claims that whether the Japanese manufacturer will stay on the grid beyond 2020 was on the agenda.
But also discussed was Red Bull's push for Honda to put even more resources into catching up with the best engines in Formula 1.
He said that is particularly important "now that Ferrari is incredibly strong in terms of the engine. Nobody can beat that," Marko is quoted as saying.
Marko's meeting with Honda also comes at a crucial time for Formula 1 more generally, with six of the ten teams - including Red Bull - currently not supportive of the 2021 rules being proposed by Liberty Media.
In Paris on Wednesday, the teams are meeting with Liberty and the FIA for the last time before the end-of-October deadline for finalisation of those 2021 regulations.
In Forbes, F1 business journalist Christian Sylt has revealed that Red Bull's annual costs have increased to a record $386 million.
In those accounts, Red Bull also laid out its concerns about the rules beyond 2020.
Red Bull stated that "changes in regulations by the FIA can have a material impact on a team's competitive capabilities with significant commercial consequences".
The accounts also said that "the company is mindful that under the new commercial and regulatory framework, post-2020, Formula 1 needs to remain attractive for Red Bull's continued support".
