War veterans back Vice President Emmerson Mnangagwa to succeed the 92-year-old president.
The African country says it needs to sell its ivory stockpile to raise money for conservation.
Zimbabwean elephants fetch around $40,000 each on the international market.
Mugabe's wife, army colleague and ex-vice president could all replace the 92-year-old.
Thousands of Morgan Tsvangirai's supporters marched through Harare on Thursday.
The president called for unity while addressing 10,000 veterans of Zimbabwe's independence war.
The influential war veterans have called on Mugabe to resign and criticized his wife Grace.
The policy obliges foreign-owned companies to give majority share to local black Zimbabweans.
The hunts were supposed to fund the construction of a football stadium in western Zimbabwe.
Veterans of Zimbabwe's independence war have indicated they want Mugabe to retire.
The El Nino weather pattern has exacerbated the country's worst drought in more than 20 years.
According to officials, the president detoured to Singapore for security reasons—but some are skeptical.
The 92 year-old leader said he was going to India on Monday and hasn't been seen since.
The world's oldest head of state isn't planning on going anywhere soon.
Joice Mujuru was sacked in December 2014 after Mugabe accused her of plotting to overthrow him.
When he finally goes, the world's oldest head of state could leave a disaster in his wake.
Airport staff noticed blood dripping from the plane when it stopped to refuel en route to South Africa.
Two former winners stepped down amid allegations they had taken nude photographs.