NEW DELHI—In a rare top-level meeting set for Thursday, senior American and Indian officials will try to push past an array of irritations in the two countries’ relationship to forge new, far-reaching agreements for military and diplomatic cooperation.

U.S. Secretary of State Mike Pompeo and Defense Secretary Jim Mattis are scheduled to meet Indian counterparts Sushma Swaraj, the country’s external affairs minister, and Defense Minister Nirmala Sitharaman to discuss ways to ramp up joint military exercises, sell more sophisticated...