After encouraging Sunni tribes in Iraq to help undermine the country’s Shiite Prime Minister Nouri al-Maliki, Saudi Arabia is now asking tribal leaders to support a new priority: fighting the insurgency led by the group that calls itself Islamic State, which the kingdom sees as an existential threat, the officials say.

These people say they hope the shift will weaken the Islamic State’s bed of support in Sunni-majority western Iraq, help cripple the insurgency and smooth the way for broader Sunni participation in a new government in Baghdad that is now in negotiation.

They also hope that the confluence of interests between Sunni-led Saudi Arabia and Shiite-led Iran in defeating the Islamic State—which stunned the region when it seized huge swaths of Iraqi territory last month—mark a first step toward easing the destructive rivalry between the two powerful countries.