"Lebanese partnership as the two countries work together to pursue common goals that advance Lebanon's sovereignty, stability, and prosperity", the USA embassy said on its Twitter page.
Washington's proposal was made by acting U.S. Assistant Secretary for Near Eastern Affairs David Satterfield, as a preview of the issues that Tillerson will discuss.
Tillerson is in Lebanon, a major recipient of US military aid, as part of a regional tour.
The heavily armed Shi'ite movement is part of the Lebanese coalition government led by Sunni Muslim politician Hariri, under a complex political system that promotes sectarian power-sharing.
"We support a free, democratic Lebanon free of influence of others, and we know that Lebanese Hezbollah is influenced by Iran".
"We have never given heavy arms to the YPG so there is none to take back", Tillerson said. He said Lebanon was sticking by its internationally recognised borders, and rejected Israeli claims over a disputed maritime zone.
The long-standing dispute resurfaced recently as Lebanon invited companies to sign exploration deals.
The council also warned against Israeli plans to build a wall along the border, saying the planned barrier would constitute a violation of both Lebanese sovereignty and UN Security Council Resolution 1701.
The grueling process, which began in 2011, will pave the way for drilling to begin in 2019 if the dispute is resolved, as Lebanon seeks to enter the hydrocarbon race in the region.
In the wake of the crisis Lebanon's political players - including Hezbollah - agreed to stick to the country's official policy of "disassociation" to stay out of regional conflicts. Prior to that, Israel had intercepted an Iranian drone which led to their jets hitting targets across Syria's territory.