Malaysia’s decades-old political rivals Mahathir Mohamad and Anwar Ibrahim set out claims to lead the Southeast Asian country on Wednesday after Mr. Mahathir’s shock resignation as Prime Minister sparked turmoil.

The struggle between Mr. Mahathir, 94, and Mr. Anwar, 72, who formed a surprise pact to win a 2018 election, has shaped Malaysian politics for more than two decades.

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Mr. Mahathir proposed a unified administration without political party allegiances. “Politics and political parties need to be put aside for now,” Mr. Mahathir said. “I propose a government that is not aligned with any party, but only prioritises the interests of the country.”

Mr. Anwar later said he opposed forming a “backdoor government” and that three parties from the former Pakatan ruling coalition had proposed his name to the king as candidate for Prime Minister. “We wait for the decision of the king,” he told a news conference.

Mr. Anwar’s Parti Keadilan Rakyat has 39 seats and alliance partners could potentially give it another 62.

While some politicians have openly voiced support for Mr. Mahathir to stay in office, it was not clear whether enough of them would give him their backing.

A unity government cutting across party lines could give Mr. Mahathir greater authority than during a spell as PM from 1981 until his retirement in 2003. But the idea was rejected on Tuesday by an alliance of four parties, including the UMNO, which ruled Malaysia for six decades until being defeated by Mr. Mahathir’s coalition in 2018.