He will be released by Leeds for England internationals, starting with next month's friendly against Australia at Upton Park.
Eriksson told the FA's website, www.theFA.com: "Brian is a very good coach, first and foremost. He already has international experience and that made him an ideal candidate.
"Secondly, he knows more than half of the players in the squad and is very popular among the players.
"Naturally, we had some other names to consider but, talking to a lot of people, including some of the players, and listening to what they had to say, he became the obvious choice and I am very happy about that."