Some MPs and parties have expressed support for the military intervention
Demonstrations have been taking place in Mauritania both for and against the coup that overthrew the country's first democratically elected president.
Police in the capital, Nouakchott, broke up a protest by hundreds of people against the coup leader, General Mohamed Ould Abdelaziz.
Earlier, about 1,000 people had marched through the capital chanting the general's name.
The African Union has demanded the president's release.
President Sidi Ould Cheikh Abdallahi was detained by renegade soldiers on Wednesday after he tried to dismiss four senior army officers - including Mr Abdelaziz, the head of the presidential guard.
Prime Minister Yahia Ould Ahmed El-Waqef - who the coup leaders had also detained - was reported to have been taken to a barracks near the presidency.
The whereabouts of the president are still unclear.
His daughter, Amal Cheikh Abdallahi, said she did not know where her father was.
"I'm very worried about his health and his security," she told the BBC's Focus on Africa programme.
"He doesn't have the right to move or to call. He doesn't have a phone. He doesn't have liberty," she said.
Meanwhile, the country's new military council said in a statement that elections would be held as soon as possible and would be "free and transparent".
Supporters of the coup march through Nouakchott
The council said it would "supervise the holding of presidential elections enabling the relaunch of the democratic process in the country".
It added that the council would "engage in a dialogue with all the political parties and all civic institutions in organizing these elections".
One MP who supported the coup said the military council would govern for six months before staging polls.
Mauritania has a long history of coups, with the military involved in nearly every government since its independence from France in 1960.
Presidential elections held in 2007 ended a two-year period of military rule - the product of a military coup in 2005.
The elections were deemed to have been free and fair and appeared to herald a new era of democracy.
Wednesday's coup drew widespread condemnation from the international community, with the EU saying it could suspend aid to Mauritania.
But in Mauritania itself, many MPs and political parties have expressed support for the military intervention, saying the civilian government had lost its way.
An opposition leader, Mohammed Sidi Ali Francois, told the BBC that the coup was regrettable but justified.
"I know absolutely we just had those elections," he said.
"A lot of money was spent and people invested a lot of time, but you know there are some situations when something has to happen and we have to make sacrifices."