ARAR, at the Saudi Arabia-Iraq Border—Saudi Arabia, with U.S. assistance, is pushing aside years of rancor with its neighbor Iraq and mounting a broad effort to win Baghdad’s allegiance and dilute Iran’s influence over the pivotal U.S. ally.

Saudi authorities are courting Iraq’s Shiite leaders, expanding the kingdom’s diplomatic presence, opening direct flights and reopening crossings closed for decades on the heavily fortified, 600-mile border.