Anwar and Mahathir, former allies and then implacable foes, joined hands to contest this week's election and oust the administration of Najib Razak.
At 92, Mahathir is the world's oldest elected leader.
The party is in the alliance that won power at the hard-fought poll.
He didn't say how long the process will take and whether Anwar could be freed before his sentence ends June 8.
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Malaysia's political landscape has been shaped in part for almost two decades by a feud between Mahathir Mohamad and Anwar, but in a striking turn of events the two leaders have joined forces.
"It is going to be a full pardon, which means not only pardoned, but he is released immediately and after that he will be free to participate fully in politics", Dr Mahathir told reporters a day after he was sworn to office following his alliance's shock win in Wednesday's election. "The king has indicated he is willing to pardon Datuk Sri Anwar immediately", Mahathir told a press conference, using a Malay honorific to refer to Anwar.
When asked if Anwar would be getting a full pardon, she answered "Yes". "He should be released immediately when he is pardoned", Mahathir added.
It was not clear when Mr. Ibrahim, who is now in hospital for treatment to his shoulder, would be freed.
"The new government vows to uphold democracy, justice and human rights for all", said the statement, delivered by the vice-president of his party, Tian Chua, who was visiting the leader in hospital. He indicated this could include Attorney General Mohamed Apandi Ali, who was appointed after Najib sacked the then-attorney general in 2015 to escape possible prosecution over a corruption scandal involving the 1MDB state fund.