William Burns met in prison with Khairat el-Shater, powerful deputy head of the group.
Officials promise the Brotherhood a political role if they comply.
Authorities announced plans Friday to besiege two sit-ins for deposed President Mohammed Morsi, but hundreds of loyalists defiantly set up a third camp near Cairo's international airport.
Said encampments of Morsi backers posed threat to national security, were "terrorizing" citizens.
Europe's top diplomat held talks with the rival sides to try to mediate an end to the country's increasingly bloody crisis.
Political reconciliation in divided nation ever further out of reach.
Most senior official to visit Egypt since ouster of Morsi insists Americans want the country's "second chance" at democracy to succeed.
Syrians who took refuge in Egypt from the civil war at home have now found themselves targets of hate speech and intimidation.
Security agencies have jailed five leaders of the Islamist group.
Mohammed Badie and nine others are charged with instigating violence that left more than 50 Brotherhood supporters dead.
