"Yes, tyrants, we're the ones who killed Chokri Belaid and Mohamed Brahmi," militant Abou Mouqatel was cited as saying.
Mouqatel, who, according to the news outlet, is wanted for murder in Tunisia, made the statement in a video distributed on social media. In the video, he appeared alongside a number of other militants, claiming to be in an IS-controlled area.
"We are going to come back and kill several of you. You will not have a quiet life until Tunisia implements Islamic law," another militant named Abou Bakr al-Hakim was quoted as saying by Al Arabiya.
Brahmi, another leading left-wing opposition figure and founder of the People's Movement party, was murdered in July 2013, sparking a new wave of protests.
Several thousand Tunisians have reportedly traveled to Syria and Iraq to fight alongside Islamic State militants, who have seized vast areas in both Iraq and Syria and declared an Islamic caliphate on the territories under their control.