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.
Welsh Labour ministers claim that alleged underfunding of Wales by the UK Treasury should be addressed first, while some Tory MPs say their party should keep its promise to voters to hold a referendum.

The "marijuana resort" would have been the first of its kind, but concerns that the stockpile could be confiscated led tribe leaders to incinerate it.
The tribe says it wants to show cooperation with federal officials for future negotiations.
Whether the tribe can sell marijuana to non-Indians has come under question.
Representatives for the US Department of Justice did not tell Santee Sioux tribe leaders a raid was imminent, but the government reserves a right to conduct a raid at any time and the tribe would risk one if all the concerns about the resort were not addressed.
The Justice Department decided last year to let tribes grow marijuana on their land.
Also in question is the origin of the marijuana seeds used for the tribe's crop.
The tribe had planned on opening a lounge selling marijuana - for consumption only in the lounge - on New Year's Eve.
"We just felt it would be best to go in with a clean slate to look for answers on how to proceed so that all sides are comfortable with it," said Flandreau Santee Sioux President Anthony Reider, adding that burning the crop was "in the best interest of both tribal and non-tribal members."
The 400-person tribe, which already runs a casino, hotel and ranch, had predicted the marijuana sales would make up to $2m (Â£1.3m) per month.
The lounge would have had games, food and a bar, and eventually a live music venue and slot machines.
In September Mr Reider said he wanted it to be "an adult playground".

The man was airlifted to Royal Stoke University Hospital in Staffordshire after the collision on an unnamed road in Tanlan, Ffynnongroyw on Monday.
The bus driver and passengers were not hurt.
North Wales Police and the Health and Safety Executive are investigating the crash.
The Welsh Assembly could be renamed the Welsh Parliament before AMs have the legal right to make the change.
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Fearing a federal raid, a South Dakota Native American tribe is burning its marijuana crop, which it had planned to sell in a resort on its land.
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