Karayev is believed to have been killed by a sniper as he got into his car after leaving his home to go to work on November 26. Although he was rushed to hospital by his bodyguards, a health official later said he had died from a bullet wound to the heart.
"We have already conducted 70 searches and detained five people," the law enforcement source said.
A police source earlier said about the attack, "There was a single bullet hole in the door of his vehicle. Presumably it was a single sniper shot which proved fatal."
On November 27, an Islamist group claimed responsibility for the mayor's murder on a website believed to have links to Chechen militants.
"Vitaly Karayev, this enemy of Allah, was eliminated near his home while on his way to city hall," the Kataib al-Khoul group said in a statement posted on the kavkazcenter site.
The group accused him of "sanctioning the persecution of women in Islamic clothes in public places, which resulted in women being repeatedly insulted and humiliated."
