Egypt could have some political appeal to the Syrian regime, because Egypt (under Nasser) was among the original three countries to join up for a “Pan-Arab” political force (along with Syrian and Iraqi Ba’athists).
Egypt has long been the military superpower of the Arab world though. They are the only Arab force that has a chance of standing up to the Turkish Army (the second biggest in NATO), which has been interfering in Syria.
Egypt is also the only secular government that might be willing to commit a large enough force in Syria to stabilize the place after a political settlement.
Egypt has a good case to get the stabilization and reconstruction job, as a party which was not involved in the conflict, speaks the language, and has a large enough force. They are also hard up economically, and could really use the job, if the International community would pay the tab.
Egypt and Syria used to be the same country, too.