President Robert Mugabe is said to be under house arrest but the whereabouts of his wife Grace, who was bidding to succeed him as president, are unknown.
South African ministers have been in the capital Harare meeting the army and political parties.
The Southern African Development Community (SADC) regional bloc will hold emergency talks on Thursday.
President Mugabe, 93, has been in control of Zimbabwe since it gained independence from Britain in 1980.
But the power struggle over who might succeed him, between Mrs Mugabe and her rival former vice-president Emmerson Mnangagwa, has split the ruling Zanu-PF party in recent months.
Last week, Mr Mugabe came down in favour of his wife, sacking Mr Mnangagwa, a veteran of Zimbabwe’s anti-colonial struggle and of Zanu-PF.
That proved too much for military leaders, who seized control of the country on Wednesday.
What’s happening with the Mugabes?
On Wednesday the office of South African President Jacob Zuma said he had spoken by phone to Mr Mugabe who had indicated “he was confined to his home but said that he was fine”.
A Catholic priest close to the Mugabe family, Father Fidelis Mukonori, is said to be involved with negotiations between Mr Mugabe and the military.
But Mr Mugabe is insisting he remains Zimbabwe’s legitimate ruler, should serve out his term and is resisting Father Mukonori’s efforts to secure him a graceful exit, Reuters news agency reported on Thursday.
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.
Grace Mugabe’s whereabouts are unclear. Namibian officials have denied reports she is there.
What about Grace Mugabe’s supporters?
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 supportive of the first lady have been detained.
They are said to include Zanu-PF political commissar Saviour Kasukuwere, Higher Education minister Jonathan Moyo and Finance Minister Ignatius Chombo.
What do the opposition parties want?
One Zimbabwean opposition leader, Tendai Biti, told the BBC he wanted to see a transitional authority in place.
“It is urgent that we go back to democracy. It is urgent 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,” he said.
He later told Reuters that he would join a national unity government if Morgan Tsvangirai, head of the Movement for Democratic Change-Tsvangirai (MDC-T) party, was also in it. Mr Tsvangirai has been abroad receiving treatment for cancer.
What about regional groups?
Mr Biti was among those calling for the intervention of regional bodies such as the SADC, which has called an emergency meeting in Gaborone, Botswana for 15:00 local time (13:00 GMT).
He also said key regional bloc the African Union (AU) could play a role.
Will President Mugabe play along?
No-one is exactly celebrating here – too much remains unknown. But shops and businesses are working and Zimbabwe is quietly adapting to the stunning fact that President Robert Mugabe is still under house arrest, and that no-one is coming to rescue him.
Instead, those now in control here – soldiers and civilians – are trying to convince the world that there wasn’t really a military coup at all. More of a forceful reshuffle.
But to win that argument, they need Mr Mugabe to play along.
So the key question now is whether the 93-year-old can be persuaded to make some sort of statement, legitimising the upheavals of the past two days, in return for, what? A peaceful retirement, the safety of his wife and family? Zimbabwe’s drama is still unfolding.
Was this a popular uprising?
Not so far, no.
There have been no reports of unrest in Zimbabwe. 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.
On national TV, Maj Gen Sibusiso Moyo denied there had been a coup, saying: “This is not a military takeover of government.”
He said Mr Mugabe and his family were “safe and sound and their security is guaranteed”.
But despite the reassurances, the AU said the power takeover and detention of President Mugabe “seemed like a coup”.
BBC contributed to this report.