Until his early release, he had been scheduled to be freed in early June.
Mr Anwar, who was formerly Dr Mahathir's political rival, has been promised leadership of Malaysia's new governing coalition when Dr Mahathir, 92, decides to step down.
Anwar served as Dr Mahathir's deputy and finance minister before being removed from his post and incarcerated on sodomy and corruption charges, which human rights groups and foreign governments criticised as being politically motivated.
Anwar emerged from the Cheras Rehabilitation Hospital on Wednesday after Malaysia's Pardon Board, which included the current King, chose to overturn the 2015 conviction, which is widely accepted to be politically motivated.
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Malaysia's king, Sultan Muhammad V, officially pardoned Anwar on Wednesday after a meeting at the palace with members of the pardons board and Prime Minister Mahathir Mohamad.
Officials in Anwar's political party said he was headed to an audience with Malaysia's king and was expected to address the public in the afternoon followed by a political rally in the evening.
When asked whether Anwar should take over as prime minister, Izzah said she believed the country deserved the best prime minister.
Anwar was heir-apparent to the premiership until Mahathir sacked him in 1998 and he was subsequently jailed for sodomy and abuse of power. He was previously the deputy prime minister under Mahatir, who led the country for 22 years until 2003.
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Once seen as a potential future leader, he had been jailed on charges of sodomy and corruption after falling out with the government.
Mahathir has earlier said he needed at least two years to fix the many problems left behind by former prime minister Datuk Seri Najib Tun Razak.
The Sydney Morning Herald said Anwar expressed confidence that he would be able to work with Dr Mahathir, whose star power and tireless campaigning had led Pakatan Harapan to victory on May 9.
Anwar must first win a seat in Parliament.
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In an ominous sign for the former prime minister, Najib and his wife Rosmah were slapped with a travel ban by Malaysia's Immigration Department on Saturday, which stopped the couple from leaving for Jakarta.