Envoys from the the Southern African Development Community (Sadc) are trying to reach a deal on the future of Zimbabwe and the man who has led the country for 37 years.
Mr Mugabe, 93, was put under house arrest on Wednesday after the army moved to take control.
Zimbabwe’s opposition leader Morgan Tsvangirai says Mr Mugabe must resign.
Sources suggest Mr Mugabe may be resisting pressure to step down, insisting he remains the legitimate president.
President Mugabe has been in control of Zimbabwe since it threw off white minority rule in 1980.
However, the power struggle over who might succeed him, between his wife Grace Mugabe and her rival former Vice-President Emmerson Mnangagwa, has split the ruling Zanu-PF party in recent months.
ast week, Mr Mugabe came down in favour of his wife, sacking Mr Mnangagwa, a veteran of Zimbabwe’s anti-colonial struggle.
That proved too much for military leaders, who seized control of the country on Wednesday.
The head of the African Union, Guinean President Alpha Condé, has warned the AU “will in no case accept” the military seizure of power. He said he was “inviting the army to return to its barracks and return to constitutional order”.
So what’s going on in Harare now?
The capital has been on edge.
A Roman Catholic priest known to Mr Mugabe for years, Father Fidelis Mukonori, is trying to mediate a deal on his future with the military.
South African Defence Minister Nosiviwe Maphisa-Nqakula and State Security Minister Bongani Bongo are meeting Mr Mugabe on behalf of Sadc, which South Africa currently leads.
Sticking points are said to include what role Mr Mnangagwa will play and the security of Mr Mugabe’s family.
Zanu-PF’s UK representative, Nick Mangwana, has suggested to the BBC that Mr Mugabe could remain nominally in power until the party congress in December, when Mr Mnangagwa would be formally installed as party and national leader.
Reuters news agency is quoting a source as saying Mr Mugabe’s wife Grace is in Mr Mugabe’s compound, along with senior figures from the “Generation-40” group supportive of the first lady – cabinet ministers Jonathan Moyo and Saviour Kasukuwere.
Mrs Mugabe had previously been thought to have fled to Namibia.
What do the South Africans want to achieve?
South Africa is hosting millions of Zimbabweans who fled after the country’s economy crashed in 2008. It has a special interest in seeing stability restored.
The Sadc mission will be pushing for a democratic solution. The body, which represents 16 countries, does not support coup-led governments as this would set a dangerous precedent in the largely peaceful region, says the BBC’s Pumza Fihlani in Johannesburg.
A Sadc emergency meeting was scheduled to take place in neighbouring Botswana to try to help find a resolution.
And Zimbabwe’s opposition?
Mr Tsvangirai, head of the Movement for Democratic Change-Tsvangirai (MDC-T) party and the main opposition leader in Zimbabwe, said on Thursday: “In the interests of the people, Mr Robert Mugabe must resign… immediately”.
Mr Tsvangirai, who has been abroad receiving treatment for cancer, also called for a “negotiated all-inclusive transitional mechanism” that would lead to “comprehensive reforms for free and fair elections to be held”.
This has been echoed by another Zimbabwean opposition leader, Tendai Biti, who told the BBC: “It is urgent that we go back to democracy… that we go back to legitimacy but we need a transitional period and I think, I hope, that dialogue can now be opened between the army and Zimbabweans.”
Reports suggest that the military are now trying to quash the threat posed by Mrs Mugabe and her allies.
On Wednesday, one of her key allies, Zanu-PF youth wing leader Kudzai Chipanga, made a televised apology for criticising the head of the army as a war of words raged prior to the military takeover.
Mr Chipanga is thought to be in army custody but insisted his statement was voluntary.
Local media reports say a number of other senior members of the “Generation-40” group have also been detained.
What are Zimbabweans being told?
Zimbabwe’s media usually toe the government line and today’s lead stories make it clear there is a new line to follow.
There is a striking absence of tough questions about what the army is doing.
“Business as usual countrywide,” says The Herald, a government-owned newspaper. Yesterday it reassured readers there was “No military takeover”.
State TV and radio stations have returned to regular programming, with Thursday’s lunchtime news bulletin on state TV giving little indication of the political upheaval.
Some privately owned newspapers have dared to address the possible end of Robert Mugabe’s rule.
“Transitional govt planned … as Mugabe cornered,” the Financial Gazette reports. “Zimbabwe scents the end of an era,” it added.
Was this a popular uprising?
Not so far, no. There have been no reports of unrest in Zimbabwe, and so far this appears to be a struggle within Zanu-PF.
Correspondents say many people have accepted that President Mugabe is being eased from office. Streets in Harare are said to be quieter than usual but people are going about their business.
On Wednesday, troops and armoured vehicles encircled parliament and other key buildings.
Hours earlier, soldiers took over the headquarters of national broadcaster ZBC and issued a statement saying that the military was targeting “criminals” around President Mugabe.