MPs from the Democrat and Pheu Thai parties are calling for a panel to rewrite the charter, which they say will make for a better and more honest democracy.
The calls were made towards the end of a two-day joint session of parliament convened to strip the Senate of its right to co-elect a prime minister.
Lawmakers were expected to vote Wednesday evening on a constitutional amendment that would see the Senate's ability to select a prime minister removed.
In the debate, MPs from the main opposition Pheu Thai along with lawmakers from the Democrat Party, a part of the coalition government, asked parliamentarians to consent to the formation of a panel to rewrite the charter.
Chawalit Wichayasut, a Pheu Thai MP for Nakhon Phanom, said he was prepared to seek the House's approval to set up a constitutional drafting assembly before parliament's term expires in March next year.
Chinnaworn Bunyakiat, a Democrat MP for Nakhon Si Thammarat, said that while he supports the charter amendment bills which are aimed at ridding the Senate of its power to co-elect a prime minister, he seconded the call for a charter rewrite body.
He said the Democrat Party had tried to push for the redrafting of the charter, but the Constitutional Court ruled that doing so would require a referendum to be organised first, to allow people to decide whether they want the charter to be reworked.
Mr Chinnaworn said the current attempts at amending the charter -- which focuses on the Senate's power to join MPs in choosing a prime minister -- do not go far enough.
However, he admitted that writing a charter from scratch given the limited time parliament has before its term expires would be a tough call. Parliament will go into recess on Sept 18 before it reopens in November.
Nonetheless, Mr Chinnaworn said meaningful changes to the constitution would ease social and political unrest which have plagued the nation.
Meanwhile, Sen Wanchai Sornsiri threw his support behind the proposed amendment to Section 272 of the constitution.
He said lawmakers should not feel too attached to the section, saying the government's functions depend on MPs, not senators.