U.S. Secretary of State Rex Tillerson on Thursday said there are "constructive" discussions underway on how to break a "stalemate" between Lebanon and Israel over gas drilling rights along the countries' disputed maritime border and maintain calm along the volatile frontier.

Tillerson was in Lebanon on a brief stopover amid a growing dispute between Lebanon and its southern neighbor, Israel, over oil and gas reserves, and Israel's construction of a border wall that Lebanon says encroaches on its territory. The U.S. has been trying to mediate in the dispute, and Tillerson suggested Israel should stop building a border wall until the border between the two countries is agreed on.

"We've asked no one to give up anything. Rather, we're looking for a solution," Tillerson told reporters at a joint press conference with Lebanese Prime Minister Saad Hariri in Beirut.