BAGHDAD—Iraqi Shiite lawmakers and militia leaders are urging Russia to launch airstrikes on Islamic State militants in their country, an escalation that would heighten tensions with Washington and increase risks of a clash between the two powers.

Since Moscow began bombing opponents of the regime in Syria last week, Iraq’s Shiite politicians, who dominate government, have been largely united in their praise of Moscow’s intervention and in calls for Prime Minister Haider al-Abadi to invite Russia to join the battle. Mr. Abadi...