Frictions burst into the open earlier this year when the Dutch government unexpectedly raised its stake in the group to almost the same level as the French state.
The surprise move followed years of disagreements over the profitability of Air France and KLM, which merged in 2004 but continue to operate largely separately.
"Yes, there may be some tensions," Air France-KLM's Canadian CEO Ben Smith told a press conference in a hangar at Amsterdam's Schiphol Airport.
"I think that when you have two airlines in a group that are so important for each country, and mean so much in each country and in the history of each country... it's only natural to see that each government wants to secure their interests for the future."
Smith added that the French and Dutch governments "are looking to ensure both companies a strong future."
In February, the Dutch government caused a political storm when it unexpectedly lifted its stake in the airline to 14 percent, just shy of the 14.3 percent held by France.
The move was prompted by doubts over the alliance's growth strategy, and worries that Dutch interests were being neglected while Air France pilots and crews were resisting the hard choices needed to streamline operations.
A series of strikes at Air France earlier this year forced flight cancellations over several months.
KLM chief Pieter Elbers said it was a good step that Paris and The Hague were in talks about the issue.