"The meeting discussed the current political situation in Egypt and its impact on the democratic transformation process," the army's official spokesman wrote on his Facebook page.
"The meeting agreed on renouncing violence, creating an atmosphere condusive to stability and national reconciliation with the participation of all trends and political forces in light of the roadmap."
On July 3, the powerful army announced a roadmap which included ousting President Mohamed Morsi, the country's first democratically elected president, suspending the constitution and installing Adly Mansour, the head of the constitutional court, as interim president.
Ever since, Morsi supporters have been staging daily mass demonstrations and sit-ins to defend his democratic legitimacy and demand his reinstatement.
Burns met Saturday with Egyptian Vice-President for International Relations Mohamed ElBaradei.
It came shortly after Burns met with leaders of the National Alliance for the Defense of Legitimacy, a coalition of Islamist parties supportive of Morsi.
The senior US diplomat also met with Foreign Minister Nabil Fahmi to discuss recent developments in Egypt and bilateral relations.