Muammer Guler, the governor of Istanbul, was quoted by the Anatolia news agency as saying that three people have been taken into custody in connection with Dink's murder on Friday.
In previous interviews with the press, Dink had cried as he talked about some of his fellow countrymen's hatred for him, saying he could not stay in a country where he was unwanted.
Turkey's private NTV television earlier said police were searching for the suspected murderer, believed to be a teenager wearing a white hat and a denim jacket.
Dink's body could be seen covered with a white sheet in front of the newspaper's entrance on Friday.
NTV said four empty shell casings were found on the ground and that he was killed by two bullets to the head.
Recep Tayyip Erdogan, Turkey's prime minister, said the killing was an attack against Turkey's unity.
"I stress that the attack on Dink is an attack on us all - on our unity, our integrity, our peace and stability," Erdogan told reporters at a hastily called news conference.
"This is an attack against freedom of thought and our democratic way of life."
Thousands of Turks took to the streets to protest against the murder.
Some 2,000 people gathered at Taksim Square, the main business and entertainment centre of Istanbul, as hundreds of riot police looked on.
"We are all Armenians, We are all Hrant Dink," chanted the protesters, many carrying red carnations and pictures of Dink with the inscription "My dear brother" in Turkish, Armenian and English.
One demonstrator waved Turkish and Armenian flags.
The protesters were marching to the offices of Dink's newspaper in the Sisli district, about three kilometers away, where hundreds of others have kept a vigil since his murder.
In Ankara, about 700 people - trade unionists and human rights activists - held a 30-minute sit-in in central Kizilay square, the Anatolia news agency reported.