Barnawi, who was previously spokesman for Boko Haram, replaced Abubakar Shekau, IS said in the latest issue of its weekly Nabaa magazine.
It did not say what has happened to Shekau, who was last heard from in an audio message in August 2015, saying he was alive and was still Boko Haram leader. An IS video released in April made the same claim.
The Nigerian army has frequently tried to kill Shekau, who became the Boko Haram leader in 2009.
Under his leadership, the Islamist group swore allegiance to IS and became more radical, carrying out more killings, attacks and violent acts.
These included the April 2014 abduction of over 200 schoolgirls from the town of Chibok in Borno state, northeast Nigeria, where the group is based.
Boko Haram has lost most of the territory it controlled 18 months ago and its seven-year insurgency aimed at overthrowing Nigeria's government has left at least 20,000 people dead.
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