Prime Minister Recep Tayyip Erdogan has said there will be no tolerance of people he accuses of seeking to harm Turkey.
The demonstrators accuse Mr Erdogan of becoming increasingly authoritarian and trying to impose conservative Islamic values on a secular state, the BBC reports.
Mr Erdogan is due to meet a group of 11 people - including artists, architects and a social media specialist - whom he hopes can mediate with the protesters.
It had been suggested he would hold talks with protest organisers, but they told the BBC that they had not been approached by the prime minister - and would refuse to meet him even if they were.
In Istanbul and the capital, Ankara, thousands of lawyers turned out to protest against the detention on Tuesday of at least 40 of their colleagues who had gathered to express their opposition to the police action.
Taxis and commuters returned to Taksim Square on Wednesday, which a day earlier had been the scene of repeated, violent clashes between police and protesters - some of whom threw stones and firebombs. Many peaceful demonstrators were also caught in the clashes.
Thousands converged on the square as night fell and were repelled by water cannon, rubber bullets and tear gas. Demonstrators sought shelter nearby, including in Gezi Park, though police said they did not plan to enter the park.
The proposed redevelopment of Gezi Park sparked anger that has widened into nationwide anti-government unrest.
The Turkish Human Rights Foundation says four people have been killed, including one policeman, in protests since 1 June.
Some 5,000 protesters have been treated for injuries and the effects of tear gas, while officials say 600 police officers have also been injured.
Protests have also occurred in Ankara, with smaller demonstrations in many other cities.