PARIS (Reuters) -Some lawmakers in France's left-wing bloc said they should focus on winning the speaker's role to show they have command over parliament even though they are well short of a majority and are struggling to form a government after a snap election.

Thursday, when the newly elected National Assembly is due to convene for the first time, is the next big date in France's political saga, which was precipitated by President Emmanuel Macron's decision to call the vote.

Lawmakers are due then to elect the assembly president, equivalent to a speaker who organises the chamber's agenda and runs debates, a central role at a time when Macron is weakened, parliament is fragmented and no party or group has an absolute majority to govern the euro zone's second-biggest economy.

Some in the New Popular Front (NFP) - the hastily assembled alliance, ranging from socialists and Greens to the communist party and the hard-left France Unbowed, which unexpectedly won the July 7 election - suggested they switch focus to getting someone among their ranks appointed, since they are wrangling over a prime minister nomination.

"There is much at stake" with the election of the lower house of parliament's chair, Clémence Guetté, of France Unbowed said on Monday. "Much of what happens next will depend on that."

France Unbowed criticised its Socialist allies for what it said was a lack of flexibility in the talks over a prime minister and said it would no longer discuss that until they agree on a joint candidate to chair parliament.

"We demand that we immediately agree on a single candidacy of the New Popular Front for the presidency of the National Assembly," France Unbowed said in the statement showing the tensions within the left-wing alliance.

Even if the left secured that role and then managed to agree on a name to put forward as government chief, it was still unclear if Macron would accept appointing a premier from its ranks as any such government would lack roughly 100 seats in parliament to form a stable majority.

Macron last week urged the mainstream parties in France's hung parliament to form a coalition able to muster a "solid" majority, putting pressure on the more moderate parts of the NFP to ditch France Unbowed and join Macron's centrists.

"When the Socialists break away from France Unbowed and their delusional programme, then we can work with them," Interior Minister Gerald Darmanin told franceinfo broadcaster.

The Socialists have given no indication they might make such a move. Instead, they have said they want their leader, Olivier Faure, to become prime minister.

Meanwhile, Macron is likely on Tuesday or Wednesday to accept the resignation of the current government, led by centrist Prime Minister Gabriel Attal, so that ministers who have been elected as lawmakers can take part in the election of the assembly's head, Darmanin said

The government would stay on in a caretaker capacity until a new team is appointed.

(Reporting by Sudip Kar-Gupta, Elizabeth Pineau, Tassilo Hummel, Blandine Henault; Writing by Ingrid Melander; Editing by Angus MacSwan and Alison Williams)