As the biggest single party, Labour put group leader Shabir Pandor forward for the role at a council meeting today but opposition groups joined forces to block his election.
Coun Pandor was elected Labour group leader earlier this month, defeating predecessor David Sheard who was council leader until the local elections.
The ousting of Coun Sheard has prompted divisions within Labour which is the biggest party on the council but does not have a majority.
The refusal by opposition groups to endorse the group leader of the biggest party is unusual and raises the possibility of Labour having to reconsider its choice. Alternatively opposition groups could come together to take control.
Kirklees Council chief executive Adrian Lythgo said: “Group leaders have agreed that tomorrow they will consult their own groups on a date to come back for an extra council meeting to vote on a new council leader.
“Between now and that meeting, conversations will take place within groups to agree a potential way forward.
“The constitutional position is that in the absence of an agreed council leader, I will run the council. I will make arrangements to consult group leaders on decisions that need to be taken in that period.”