"The martyrdom-seeking brother Abu Zakariya al-Britani -- may Allah accept him -- detonated his explosives-laden vehicle on a headquarters of the Rafidhi army and its militias in Tal Kisum village, southwest of Mosul," the claim quoted by SITE said.
The IS statement did not say when the bombing occurred.
The jihadist group uses the word "rafidha", which means "rejectionists", to refer to Shiite Muslims in a derogatory way because it considers them heretics.
They are fighting alongside other Iraqi forces -- including the army and the federal police -- as part of a push that started on Sunday to retake the west bank of Mosul.
Tens of thousands of Iraqi forces launched a massive offensive on October 17 to retake the city, which is Iraq's second largest and the only remaining major stronghold of the jihadists in the country.
They retook control of the eastern side of Mosul last month.
The IS statement said that the British fighter's attack, and that of another suicide bomber of Iraqi nationality, caused many casualties but AFP could not immediately verify the claim.
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