Bersatu-splinter party Pejuang wants the MACC to probe Prime Minister Muhyiddin Yassin over Gua Musang MP Tengku Razaleigh Hamzah's claim that he was offered the Petronas adviser post.
The party's legal team said the alleged offer is a clear violation of Section 23(1) of the MACC Act which makes it an offence for any officer of a public body to use his office or position for any gratification whether for himself or an associate.
The Act defines positions as gratification.
"The prime minister's act of using his position to 'bribe' the Gua Musang MP with the Petronas chairperson (sic) is not only something that is embarrassing but is a serious bribery offence that should be investigated immediately by MACC," Pejuang's legal team said in a statement.
They added that MACC is empowered to launch investigations on their own without waiting for a report to be lodged.
Razaleigh, who is better known as Ku Li, made the claim that he was offered the Petronas adviser post in a letter to Dewan Rakyat speaker Azhar Harun that was released last week.
In the letter, the Umno veteran said he rejected the post due to concerns over Article 48(1)(c) of the Federal Constitution on "the disqualification for membership of Parliament" if "he holds any office of profit".
He said this was meant to prevent MPs from supporting or voting for a particular government and a form of political graft.
He said political graft has caused Malaysia's parliamentary democracy to face a serious challenge and he was worried whether such a practice had infiltrated the nation's democracy quietly.
After the Perikatan Nasional government came into power, the opposition had criticised the various political appointments made by Muhyiddin.
They had claimed that this was an attempt to consolidate support for himself and subsequently his razor-thin majority in the Dewan Rakyat.
Razaleigh had called on Azhar to allow a motion of no confidence against Muhyiddin to be debated in the Dewan Rakyat.