Mahathir was once Najib's mentor but he left the former ruling party over the 1MDB affair and joined the opposition.
However, Dr Mahathir, who now leads the coalition and was sworn in as prime minister on Thursday, has said it may take up to two years for him to hand over the reins as he wants to stick around to fix the problems he sees plaguing the nation.
Speaking via video conference at The Wall Street Journal CEO Council Meeting in Tokyo, Dr Mahathir, 92, said: "We are slowly getting to the bottom of things and many of our senior officers are volunteering information accompanied, of course, by documents".
Analysts previously said the National Front might lose the popular vote but hold onto a majority in parliament due to an electoral system that gives more power to rural Malays, the party's traditional supporters.
She says that Anwar's recent hospitalization might "encourage him to slow down a bit", but he is known to be eager to get back into the fray.
Differences have already appeared between supporters of Mahathir and Anwar over cabinet positions and Anwar's role.
The four parties that make up Pakatan Harapan, including Anwar's Parti Keadilan Rakyat (People's Justice Party), had agreed before the election that Mahathir will be premier and Wan Azizah will be deputy premier.More news: 'Brooklyn Nine-Nine' saved by NBC days after Fox canceled it
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Anwar has spent eight of the past 20 years in prison on politically-motivated sodomy charges, and as his one-time rival, Mahathir Mohamad, secured a remarkable election victory six days ago, he was in hospital recovering from shoulder surgery.
The head of the Malaysian Anti-Corruption Commission, who was appointed by Najib in 2016, has resigned but Mahathir said he could also be investigated.
Anwar sought to allay those concerns, saying he was not in a hurry to take over from Mahathir.
In the 1990s, Mahathir was prime minister, Anwar who worked under him was his deputy and clear heir-apparent.
Mahathir, at that time said he had to remove Anwar because he was "not fit for office". Ironically, as Anwar emerges from prison, the prospect of Najib facing prosecution looks increasingly likely. He says the funds were political donations from a Saudi royal family member.
Mahathir is also racing ahead with an investigation into graft at 1Malaysia Development Berhad (1MDB), a state fund founded by the ousted Najib that is sunk in a multi-billion-dollar graft scandal.
"If there is no case, we don't go for detention without trial", he said. Jack Seng, another supporter wearing a shirt with a logo of Anwar behind bars, said he thought Mahathir and Anwar would resolve any tensions between them. He has warned that Najib will have to "face the consequences" if found guilty of wrongdoing.