The shooting comes amid increased Egyptian efforts to block a surge of migrants from entering Israel after rumours spread that Israel would welcome them and give them work.
The source said border police had spotted the migrants walking toward the border and ordered them to stop, opening fire when they failed to do so.
Racism and abuse
Police arrested 22 of the migrants, who were from Sudan, Ivory Coast and Eritrea. But others successfully made it across the border.
While the migrants are drawn to Israel by reports of better living conditions and hopes of gaining asylum there, they are also keen to leave Egypt were activists say they face persistent racism, abuse and economic marginalisation.
The Sinai Peninsula has long been host to Bedouin smuggling rings who will slip people, weapons, or drugs and tobacco across the Egyptian frontier into Israel or the Gaza Strip for a price.
Earlier this month, Egyptian border police shot at a group of Sudanese trying to enter Israel, wounding one.
Meanwhile, last month, Egyptian authorities arrested 106 Sudanese trying to cross into Israel, Egyptian security sources said.