Bradley replaces long-standing coach Hassan Shehata who left in June after holders Egypt struggled in qualifying for the 2012 African Nations Cup finals. They have since been eliminated.
Bradley, fired by the U.S. in July, was on an Egyptian FA shortlist that included the experienced Francisco Maturana, who has managed Peru, Ecuador and his native Colombia, and Zoran Filipovic, the former Yugoslav striker and Montenegro coach.
The American will be given the task of getting Egypt to the World Cup finals for the first time since 1990.
Bradley is expected to arrive in Cairo on Sunday or Monday to sign a contract, Egyptian FA spokesman Azmi Mugahed told MENA. Details of the deal will be announced next week.
Bradley took over as the permanent head coach of the U.S. in 2007 after being taken on as interim coach when the Americans flopped at the 2006 World Cup in Germany.
He enjoyed instant success, winning the 2007 CONCACAF Gold Cup at home and leading the U.S. to the final of the 2009 Confederations Cup in South Africa.
Bradley guided the team through the qualifiers for the 2010 World Cup, also in South Africa, but they lost in the second round to Ghana after drawing with England in the group stage.
Bradley was linked with a move to English Premier League club Aston Villa but signed a four-year extension last year to stay on as U.S. head coach.
However, his position came under renewed attack after the U.S. were beaten by Panama in the group stages at this year's CONCACAF Gold Cup. They recovered to reach the final against Mexico but squandered a 2-0 lead to lose 4-2.
