In a referendum in April, a majority of the island's population voted to end Sunday flights to Rarotonga, with those in support saying Sunday is a sacred day when businesses shouldn't operate.
But tourist operators says the flights are essential, and canning them would have a severe impact on Aitutaki's tourism industry and economy.
At a news conference on Friday, Prime Minister Henry Puna said he will be meeting with Aitutaki's leaders and representatives from both sides of the issue at a meeting in Rarotonga.
Our correspondent, Flo Syme Buchanan, says Mr Puna said he was confident he had an acceptable solution.
"He didn't give any suggestion as to what that solution would be [but] he seemed quite confident that the solution would be acceptable to both parties and it raises a lot of interest as to what that solution actually is because it's two very different sides."