Ahmad Zahid Hamidi, Umno president and member of Parliament for Bagan Datuk, claims to have been told that a big chunk of Umno and Barisan Nasional MPs have voiced their support for Anwar.
Anwar said he was "approached by several MPs from different parties" who were dissatisfied with the leadership of Prime Minister Muhaidin Yassin.
The government will continue to ensure that banking institutions help Malaysians and businesses during this hard period, he said in the Kita Prihatin (We Care) special address to the nation.
Anwar said he has been granted an audience with the king Tuesday but it was postponed because the king was in a hospital for treatment.
But the premier has a razor-thin majority and there has been speculation he could seek an election to win a stronger mandate. The Malaysian ringgit and stocks fell after Anwar's comments.
He did not reveal how many lawmakers were backing him.
Malaysian Prime Minister Muhyiddin Yassin yesterday announced an additional economic stimulus package worth RM10 billion (S$3.3 billion) in a bid to help industries and citizens badly hit by the coronavirus pandemic.
"Without going through this process, Datuk Seri Anwar's statement remains a mere claim". Many Malays were unhappy with what they saw as too much focus on non-Malay interests, and particularly those of ethnic Chinese, under the Mahathir-Anwar coalition.
In a separate statement, the secretaries-general of six coalition parties within the National Alliance expressed support for Muhyiddin to stay on. He could also opt to dissolve parliament for snap federal polls.
Mahathir, whose past feud with Anwar charted the course of Malaysian politics over the past 20 years, said he will "wait to see if this is another episode of making claims that can not be substantiated", the Edge newspaper reported.
"Internal politics in Malaysia remain messy, but are unlikely to affect economic policies", said Mingze Wu, a currency trader at StoneX Group in Singapore.
Asked if former premier Dr Mahathir Mohamad would be part of his soon-to-be-formed government as claimed, Anwar said the Langkawi MP had yet to pledge support.
Muhyiddin came to power in March after securing a parliamentary majority with the support of UMNO, which was defeated in the 2018 election.
Mahathir, now 95, became prime minister for a second time and Anwar was released from jail, where he had been serving a sentence after being convicted of dubious sodomy charges.
Mr Anwar, 73, has had a tumultuous political career over the past two decades. "Until now - seven months after taking office during an unprecedented national and worldwide crisis - the Prime Minister has not held a single press conference following his regular Cabinet meeting".