The heads of state from Palau, Nauru, Kiribati, the Federated States of Micronesia and the Marshall Islands issued the warning in a communiqué earlier this month.
The statement came at the end of their Micronesian Leaders' Summit and said the five nations would leave the Forum if their nominee for the regional group's Secretary-General was not given the role.
They indicated there had been a 'gentlemen's agreement' that the next Secretary-General would be from Micronesia.
Following the Pacific Foreign Ministers Conference this week, the meeting chair, Tuvalu's Simon Kofe, said the situation was unfortunate.
"It's something that the leaders will look into. We are certainly very concerned about the threat from the Micronesians to pull out from the Forum."
Dame Meg Taylor from Papua New Guinea ends her tenure as Secretary-General in January.
Her replacement was expected to be announced next month.
Simon Kofe said the process had been a challenge but importantly, the aim was to maintain solidarity in the region.
While he did not give any details on the election process, Kofe said there had been a huge interest from member countries contesting the position.
"The process will be determined by the leaders. But in terms of dates, we are looking at November. The majority of members have expressed their preference for the first week of November."
Kofe said member states were expected to provide their options on the election of the new secretary-general before the weekend.
The position was one of two key issues discussed at this week's conference of Pacific Foreign Ministers.
Kofe said discussions also centred around the Covid-19 pandemic.