Pheu Thai MPs reiterated at their latest meeting that the next House speaker must be a candidate from their party and said the demand would be relayed to the Move Forward Party (MFP) during their talks on Wednesday.
This will also reportedly be among the key issues raised at a planned meeting of the eight prospective coalition parties on Thursday.
Pheu Thai executives and MPs held separate meetings to discuss the matter on Tuesday.
Party leader Cholnan Srikaew said a team of negotiators held talks with the MFP and proposed that the two parties get 14 cabinet seats each, with the MFP entitled to the prime minister's position and Pheu Thai taking the House speaker role.
The MFP accepted the proposal for consideration but has yet to give a definitive answer, Dr Cholnan said.
He said the party's executive committee supports the proposal, and it was presented at the meeting of party MPs on Tuesday.
Most of the MPs present agreed Pheu Thai negotiators must stand firm on the proposal at Wednesday's planned talks, he said.
Deputy Pheu Thai leader Phumtham Wechayachai said most party MPs want the negotiators to thrash the proposal out with the MFP, even though this does not represent the party's official stance.
Chusak Sirinil, another deputy leader of Pheu Thai, said the House speaker must remain neutral and be acceptable to all parties given the role they play in ensuring House sessions proceed smoothly.
"This is not a battle for the House speaker position. Pheu Thai still recognises the need for the MFP to establish a government with [its leader] Pita Limjaroenrat as prime minister.
"But we believe a House speaker who is a Pheu Thai-chosen candidate would be able to ensure the House meetings go smoothly," Mr Chusak said.
According to sources at the MFP, the party decided on Tuesday that Padipat Santiphada, a Phitsanulok MP and member of the party's executive committee, would be nominated for House speaker.
Previously three other candidates were strongly tipped for the post -- deputy party leader Natthawut Buaprathum, party-list MP Parit Wacharasindhu, and Teerajchai Phunthumas, a Bangkok MP.
MFP secretary-general Chaithawat Tulathon said earlier the new House speaker should hail from the MFP, which won the most House seats in the May 14 election.
He said the two parties would act in line with the mandate of voters and work together to form a coalition government.
According to Senate Speaker Pornpetch Wichitcholchai, His Majesty the King will preside over the state opening of parliament on July 3, and the House of Representatives will convene on July 4 to select the speaker and two deputies.
Meanwhile, Deputy Prime Minister and Palang Pracharath Party leader Prawit Wongsuwon met party executives and 40 newly endorsed MPs on Tuesday at the Foundation for the Conservation of Forests in Five Adjoining Provinces at the compound of the 1st Infantry Regiment in Bangkok.
According to sources, Gen Prawit stressed the need for all party MPs not to break ranks and to vote in the same way for both the new House speaker and prime minister.
But the attendees did not discuss who should be nominated for speaker, the sources said.
Akanat Promphan, secretary-general and a party list MP of the United Thai Nation Party (UTN) -- of which Prime Minister Prayut Chan-o-cha is the chief strategist -- said the party would not vote for the MFP's candidates for either the House speaker or prime minister.
The UTN will not support any party that seeks to amend Section 112 of the Criminal Code, or the lese majeste law, Mr Akanat said, referring to the MFP.
He said none of the outgoing coalition parties would nominate candidates for House speaker, to the best of his knowledge.
He also denied speculation that Pheu Thai would ask the UTN to vote for its candidate for that position in parliament.