Sonia Backes of the Caledonian Republicans suggested that the rival anti-independence groups in Congress choose another presidential candidate than Philippe Germain, given his repeated rejection.
Mr Germain has failed twice in two months to secure a majority among the 11 ministers making up the government when they met to elect a president.
Ms Backes said a more conciliatory person needs to be presented as a candidate, suggesting another minister Bernard Deladriere.
She said furthermore there have to be changes to the government's economic policies.
RNZI reports the Caledonian Republicans hold one seat in the collegial government while the pro-independence side has five and the groups backing Mr Philippe also have five.
There was a similar stalemate in 2015 which ended when the pro-independence side decided to help elect Mr Germain.
His government fell in August when a minister Philippe Dunoyer took up the seat he had won in the French National Assembly election.