Among the possible names in the looming battle to take over are his wife Grace, 50, and his two vice-presidents Phelekezela Mphoko and Emmerson Mnangagwa.
"There is none of us who is a better comrade than another," Mugabe told an audience of several thousand in the western resort town of Victoria Falls.
"We don't want to hear people saying 'these ones belong to Mphoko, these ones to Mnangagwa.' If all people belong to factions, who then are mine?"
"We had come to a point where there were some in the military, the police and the intelligence services joining factions. Let's stop that."
Mugabe who wore a floral yellow jacket decorated with party symbols and his image, was flanked at the top table by Grace, who wore a dress with similar motifs.
"Your ambition should not divide the people," he told the meeting of party loyalists from around the country.
"People don't belong to factions. They belong to ZANU-PF.
"We cannot succeed if we continue quarrelling and quarrelling."
Mnangagwa is widely seen as positioning himself for Mugabe's post, while former deputy president Joice Mujuru, who was ousted last year, is also a potential challenger.
Speculation has grown that Grace Mugabe also harbours ambitions to take power.
She recently said that she had no further wishes beyond being the president's wife, and said she would "put Mr Mugabe in a wheelchair" and take him to stand in elections next due in 2018.
Grace was appointed leader of the ZANU-PF powerful women's wing last year and went on a campaign that led to the expulsion of Mujuru, whom she accused of fomenting division as well as corruption and practising witchcraft.
The president, who has been accused of brutally quashing dissent for decades, has never named a preferred successor.