The inconclusive result prolongs political instability in the Southeast Asian nation, which has had three prime ministers in as many years, and risks delays to policy decisions needed to galvanise an economic recovery.
Investors reacted negatively to the outcome of Saturday's vote as the ringgit currency MYR= weakened and the Kuala Lumpur stock market .KLSE slid over 1 percent in early trade on Monday.
Anwar's multi-ethnic coalition won the most number of seats in the Saturday election with 82.
Muhyiddin's conservative Malay Muslim alliance took 73 seats, but then secured support from two smaller political blocs on Sunday, giving it control of 101 - still short of the 112 needed for a majority.
Muhyiddin's bloc includes an Islamist party that has called for sharia law and that secured the largest number of seats of any single party.
The incumbent Barisan Nasional alliance, led by the United Malays National Organisation (UMNO) party - which had long been the dominant force in Malaysia - suffered its worst ever electoral performance, but could still end up playing a crucial role by throwing its support behind either of the leading candidates.
Media reported that Anwar's coalition was meeting with Barisan on Monday morning, though Muhyiddin had been a junior partner in the outgoing government and had been a senior UMNO leader until a few years ago.
King Al-Sultan Abdullah, whose largely ceremonial role includes the power to appoint as prime minister a lawmaker he believes will command a majority, could play a decisive role.
The king has asked political parties to each present the name of a lawmaker that they think has majority by 2 p.m. (0600 GMT) on Monday.
The decline of the graft-tainted UMNO party, which had led every government since independence from British colonial rule in 1957 up until its first defeat in the 2018 election, has ushered in a new uncertain phase in Malaysia.
Massive corruption, mostly related to the looting of billions of dollars from sovereign fund 1Malaysia Development Bhd (IMDB), for which former prime minister Najib Razak was jailed earlier this year, seriously wounded UMNO's image.
But up until recently, the country has a record of being one of the most politically stable in a region that has suffered its share of military coups, violent political upheavals, and insurgencies.
It would be a remarkable turnaround if UMNO were to back Anwar, who launched an anti-corruption reform movement when he acrimoniously left the party in the late-1990s. During his time in opposition, Anwar spent a decade in jail on convictions for sodomy and corruption, which he says were politically motivated.
His multi-ethnic coalition opposed the affirmative action programme for ethnic Malays that was championed by UMNO
