ISLAMABAD, Pakistan  American officials pressed their Pakistani counterparts on Thursday to reopen a vital supply route for American and NATO forces in Afghanistan, as relations deteriorated after the fourth strike by coalition helicopters in a week killed three members of Pakistan’s border force.

Pakistan angrily closed the crossing to protest the strikes on its side of the border, leaving American officials to use meetings and phone calls to try to soothe relations and get the route reopened. Both sides indicated that they might be able to resolve the dispute with a joint investigation.

But the border closing, and the exceptional series of strikes by piloted aircraft, as opposed to drones, signaled a general increase in tensions between Pakistan and the United States, already uncomfortable allies that are pursuing competing interests in the Afghan war.

The C.I.A. carried out a record number of drone attacks inside Pakistan last month, and new reports surfaced this week of unlawful executions by the Pakistani Army in areas where it has opened operations against Taliban forces threatening the government.