Notably, Tillerson's visit comes amid mounting tensions between Lebanon and Israel.
"We support a free, democratic Lebanon free of influence of others, and we know that Lebanese Hezbollah is influenced by Iran".
Under Lebanon's constitution, the president must be a Maronite Christian, the prime minister a Sunni Muslim, and the speaker of parliament a Shia Muslim.
"Hebollah's entanglement in regional conflicts threatens the security of Lebanon", he added during his visit, the first by a U.S. Secretary of State to Lebanon since 2014.
Israel has recently escalated its threats over Lebanon's invitation for offshore gas exploration bids along the countries' maritime border claiming that Lebanon will be drilling in areas owned by Israel.
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.
Mr Aoun, a former army general, became president of Lebanon in October 2016, after a surprising shift in allegiances across the country's deeply divided political parties, ending an nearly two-year-long power vacuum.
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.
U.S. Secretary of State Rex Tillerson, who is now on a five-nation tour of the region, arrived in Beirut on Thursday for talks with Lebanese officials.
Rafic Chlela, head of President Aoun's media office, said Mr Tillerson had arrived a few minutes earlier than expected, and the meeting began on time.
At his meeting with Berri, a pro-Hezbollah politician, one of the photographers asked the men to shake hands.
Hezbollah's complicated role in Lebanon's coalition government was highlighted last November when Mr Hariri dramatically resigned from his position while on a visit to Riyadh, a decision widely believed to have been orchestrated by his party's Saudi backers in order to weaken Hezbollah.
The visit comes amid a new US effort to squeeze Iran with sanctions against Hezbollah and the Trump administration is hitting Hezbollah's financial network with new penalties as part of its efforts to limit Iran's influence throughout the region.
Turkey looks to ease Aegean tensions
He added that the two premiers agreed to ease tensions and keep communication lines open via political and diplomatic channels. Coast guard officers warned off Kammenos, who was heading to the Kardak islets in an assault boat to lay a wreath there.