This is a sign that the government is intent on meting out punishment over the massive protests that swept the country in June.
The Aksam newspaper said police had provided to Istanbul prosecutors a list of 35 names of people who had allegedly insulted Prime Minister Recep Tayyip Erdogan or other officials on Twitter or Facebook.
Deputy Prime Minister Bekir Bozdag acknowledged the probe, but would not confirm the list. It was not clear exactly what the posts said.
Prime Minister Recep Tayyip Erdogan has faced tough international criticism over his government's heavy-handed crackdown on the unprecedented demonstrations.
But in a possible attempt to soften the blow to the country's democratic reputation, his deputy also said the government would propose further checks on the country's historically powerful military.
Erdogan earlier this month branded Twitter a social "menace" for spreading "lies" after many people turned to the social networking site and Facebook for information.
Many Turkish media outlets provided little coverage in the early stages of the demonstrations, likely intimidated into self-censorship by the government's previously tough approach to journalists.