The Welsh Assembly could be renamed the Welsh Parliament before AMs have the legal right to make the change.
Naming rights are among the powers set to be devolved in the Wales Bill now going through the UK Parliament.
But Jane Hutt, Labour's chief whip in the assembly, will call for the Welsh Parliament name to be adopted "at the earliest opportunity", and used unofficially until formalised.
AMs will debate the issue in the Senedd next Tuesday.
Ms Hutt will table a motion proposing that "the National Assembly for Wales agrees that: (a) its name should be changed to the "Welsh Parliament" at the earliest opportunity; and that (b) it should be known unofficially by that name until such a name change can be formalised".
A Welsh Government spokesperson said: "The First Minister is opening up debate among Assembly Members, early in the life of this Assembly, to consider the most appropriate name for the institution.
"It is ultimately a matter for the National Assembly itself; legislation is not required to change what the institution calls itself."
Asked for a response, a Wales Office spokesman said: "The Wales Bill will give the Assembly the power to call itself what it wants."
Glyn Davies, the Conservative backbench MP for Montgomeryshire, said he would refer to the assembly as a parliament, saying on Facebook that AMs "should have done it years ago".
In May, Welsh Secretary Alun Cairns spoke of the Wales Bill offering opportunities "to deliver a real parliament for Wales, with greater powers and a government responsible for raising as well as spending money".
However, differences remain between the two governments on whether powers over income tax should be devolved without a referendum.