Iraqi security forces and their U.S. partners are divided about where to next confront Islamic State after their victory over the extremist group in Tikrit last week.

Both hope to capitalize on the momentum to reclaim all of the surrounding province of Salahudeen. The bulk of that effort will focus 25 miles northwest of the provincial capital Tikrit on the refinery city of Beiji, which is partially held by Islamic State forces.

Beyond...