The primary authority for negotiating agreements between the United States and foreign nations (host countries) rests with the U.S. Department of State, led by the Secretary of State.
While the President holds the power to make treaties (with Senate advice/consent) and executive agreements, the State Department executes the diplomacy. Other agencies (USTR for trade, DoD for defense cooperation) often lead specific negotiations but operate under State Department coordination and the President's foreign policy direction.