Eighty-six people - including former army officers, journalists, a former university dean and a lawyer - are already on trial in the case.
They have pleaded innocent and accuse the government of Recep Tayyip Erdogan, the prime minister, of attempting to silence critics.
The case is widely perceived as being part of a power struggle between Turkey's secular establishment, including parts of the military, and the democratically elected and religiously conservative government.
It also has raised concerns about political instability in Turkey, which has seen four government forced from power by the military since 1960.
Turkey's state-run Anatolia news agency said almost 40 people were detained during Wednesday's police raids, and local media said the suspects were taken into custody at homes and offices in about a dozen cities or towns.
Police confirmed the raids, but provided few details.
Deniz Baykal, the leader of the country's main opposition party, accused Erdogan's government of conducting a "political vendetta" against anyone in opposition to the government.
"The detentions have turned into a revenge campaign," Baykal said. "These are respected people who defended the secular republic."
Mehmet Ali Sahin, the justice minister, rejected Baykal's allegations, telling reporters that the arrests were "purely legal, not political".