At least 10 people were killed and 30 wounded in the attack, according to initial reports by police and hospital sources in the city, capital of Anbar province.
"At around 0400 GMT, IS launched seven attacks with suicide bombers driving Humvees in the areas of Hawz, Malaab, Toi, Albu Faraj and Albu Eitha," police major Mustafa Samir said.
For months, the government has been fighting off IS militants who control most neighbourhoods on the outskirts of the city and regularly attack security forces in more central districts.
Pro-IS accounts on social media said a Belgian, a Syrian and a militant from the Caucasus were among the suicide bombers.