Lebanon has vehemently denied those claims, arguing that the wall encroaches on Lebanese territory in 13 areas along the blue line which demarcated Israel's withdrawal from southern Lebanon in 2000, with Israel's recent incursions suggesting that it views the blue line as the definitive border between both sets of countries. On Wednesday, speaking in Amman, Tillerson had appearing to soften Washington's tone, conceding that Hezbollah is part of the "political process" in Lebanon.
He spoke Thursday at a joint press conference with Lebanese Prime Minister Saad Hariri, an ally of the West who is at odds with Hezbollah.
Tillerson's visit comes at a time of tension between Lebanon and Israel - which fought a devastating war with Hezbollah in 2006 - over issues such as offshore energy exploration.
Lebanon's president says he has asked visiting U.S. Secretary of State Rex Tillerson to work on preventing ongoing Israeli violations of its sovereignty. "Hezbollah entanglement in regional conflicts threatens the security of Lebanon and has destabilising effects on the region", he said of the Iran-backed Shia movement, which is part of the Lebanese government.
"Mr President, thank you for the warm welcome and the open, frank, and productive discussion", Tillerson wrote in the Lebanese palace's visitors book after the talks.
Lebanon is also protesting Israeli violations of Israeli airspace, often to bomb targets inside neighboring Syria.
Monday's meeting, according to governmental sources, "looked into acting assistant US secretary of state David Satterfield's proposal" to resolve a Lebanese-Israeli dispute over an 860-sq km maritime area.
Israel regards Hezbollah as the biggest direct threat on its borders and has repeatedly struck it in Syria, where the group is fighting alongside President Bashar al-Assad in a civil war now entering its eighth year.
The long-standing border dispute between the two countries resurfaced recently as Lebanon invited foreign companies to sign exploration deals.